For the reduced simplicial set whose simplices are subgraphs of a finite labelled undirected multigraph equipped with ordered partitions of their vertices, we construct explicit mod-two cohomology classes indexed by subgraphs. Vertex restriction pulls a supported top-dimensional cochain into a global cocycle. Permutation cycles pair with these classes as a Kronecker basis, while a vertex filtration bounds the total cohomology dimension using augmented Boolean order complexes. Alexander-Whitney multiplication recovers the graph Hall product. This gives a candidate proof of the finite-graph Hall/cohomology identification stated as Conjecture 1.1 by Toury. The construction concerns labelled subgraphs, not isomorphism classes of rooted forests. Loops and labelled parallel edges are allowed. Exact finite checks accompany the proof; priority and external mathematical assessment remain pending.
Working paper. Citable, but not yet screened for the AIRR collection.
On finite periodic unit-spacing lattices, necessary coarse-information budgets constrain flat Hodge forms and constructed derivative repairs. We consolidate physical Fourier-band, tangent, exact regular-sector, signed-weight and full-volume rank results with separately pinned formal evidence and analytic arguments. A critical inverse-conductance moment suffices for a global spectral observation of the mean-budget order. Actual rectangular means have a general D/4 upper rate and the target D/2 rate for an explicitly restricted scaled-cube class. Constructive counterexamples obstruct a retained-grid single-threshold algorithm and a universal exact supplied-cube decomposition, while the same profiles admit successful rectangular partitions. The unrestricted critical rectangular question remains open. No interacting or continuum Yang-Mills mass gap is established.
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Access to technology can improve performance during practice without producing competence that persists when assistance is removed. This article examines a more specific hypothesis than the general superiority of technology: instructional designs that preserve independent attempts while supporting feedback, retrieval and subsequent application may increase durable independent learning within a fixed study period. A critical narrative synthesis of thirteen research publications, supplemented by two contextual sources on El Salvador, brings together research on learning, computer tutoring, generative artificial intelligence and chess. Findings include benefits, null effects and harms. They do not establish a universal technological advantage or isolate technology as the cause of historical improvements in chess. The proposed framework distinguishes the organization of practice from the cognitive operations learners are intended to acquire. It operationalizes an observable attempt, feedback and reapplication cycle without treating cycle frequency as evidence of learning by itself. A proposed four-arm randomized trial compares active print study, fixed digital practice, constrained adaptive digital practice and direct access to solutions, with common core blocks protecting scheduled syllabus coverage. The primary outcome is unaided performance on novel chess positions four weeks after training. Improved assisted performance without a delayed advantage would not establish durable learning. The contribution is a falsifiable research framework and a proposed protocol, rather than a new empirical finding or a registered study.
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Taking local matrix adjoints along a forward random process does not reverse its temporal order. We give finite, ergodic qubit examples in which first and eventual Kraus-span injectivity lengths change under this operation. The failure persists for bistochastic, two-sided faithful local channels; a three-phase example also changes the onset distribution. These examples contradict the pointwise adjoint-length identity in Lemma A.1 of arXiv:2607.01446v1, without providing a counterexample to that manuscript's main parent-Hamiltonian or spectral-gap theorems. We state a corrected inverse-base identity. It preserves fixed-block distributions, and, under local and dual faithfulness, preserves first and eventual onset laws. We also replace an invalid common-kernel step in the source appendix by a Choi-rank convergence argument, recovering its eventual-strict-positivity/injectivity implication under the cited ergodic convergence theorem. The required normalization and endpoint shift are explicit. Fixed-block, first-occurrence, eventual, and pathwise assertions remain distinct.
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For a channel on a d-dimensional system with ordinary trace-distance contraction at most r, we give a common-component certificate after an explicitly specified measurable preparation instrument. Its mass is at least max{0, 1-(d-1)r}, independently of the outcome cardinality. The same certificate yields a completely positive replacement component after the associated measure-and-prepare channel. We determine the worst-case envelope of this positive component over the stated class using classical exclusion channels. We separate these assertions from optimal complete contraction and from mixing of arbitrary observed instruments. Two elementary examples show why the recorded instrument and the topology on posterior laws must be retained. The measurement coefficient, replacement framework, and downstream contraction principles are established prior results; worldwide novelty of the dimension comparison is not asserted.
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For a finite classical-quantum channel mapping i to tau_i, we prove that its complete equal-reference Umegaki contraction coefficient is one exactly when the intersection of the prepared-state supports is zero. The known positive-replacer bound supplies strict contraction in the complementary case. An operator-valued simplex factorization and a normalized graph projector give the converse with faithful separable hypotheses, uniform classical input, reference dimension equal to the input alphabet size and a joint binary readout. For every fixed channel, input, output and measured relative entropies tend to log 2. Two nonorthogonal pure outputs already exhibit the phenomenon: a certified finite binary retention exceeds the complete trace-distance coefficient, so a Hermitian Doeblin or trace bound cannot be substituted for the positive-replacer Umegaki bound. The graph effect can be nonseparable, and its convergence is not uniform over prepared states approaching equality. For classical outputs, a separate weight-independent Schur argument yields uniform convergence over all zero-Doeblin channels at fixed input size. We also determine exact SLD, minimal chi-square, triangular, BKM and Umegaki coefficients for symmetric classical channels, and distinguish binary input support from binary hypotheses or readout. Exponential conditioning, rare events, adaptive measurements and the quantum input denominator delimit the result. No capacity advantage, robust implementation, minimal reference dimension, general LOCC realization or worldwide priority is claimed.
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We determine the complete equal-reference contraction coefficients of the minimal quantum Fisher metric, its quadratic chi-square divergence and the finite triangular Petz divergence for every unital qubit channel. For a faithful normalized Choi state J they equal 1-16/Tr(J inverse); for singular J they equal one. A qubit-qutrit lower family matches a harmonic-weight upper and gives an exact stabilization envelope, including a maximal factor of nine between coefficient suprema. For dimension d depolarization, a commuting Bell family matches lambda^2 kappa/[1+lambda(kappa-1)] for each normalized nonaffine operator-convex Petz generator up to kappa=d^2/2. A principal-rich rank-two ladder extends SLD and triangular sharpness to kappa=d(2d-1)/2, with actual-marginal budgets and reference dimension 2d-1. A separate logarithmic calculation yields the complete equal-reference BKM and finite Umegaki lower lambda^2 P/[1+lambda(P-1)], P=d(d-1), with a constrained profile and four interval-certified finite comparisons. We also identify qubit classical-reference and separable/PPT restrictions and give a localized Doeblin counterexample. These results do not determine unrestricted BKM, Umegaki, actual-mutual-information or capacity optima; the earlier ladder lower is not a unitary phase model. Faithful limits, prior dependencies, adverse controls and review-context limits are explicit. A separate norm-one local Hamiltonian with reference dimension d realizes the BKM/Umegaki lower at fixed finite phase. Certified relative-entropy pairs exceed the common-Bell-diagonal class at both noise signs. The exact classical benchmark is reconstructed throughout the physical noise range, without a classical priority claim. The phase family spends order approaching d squared and loses absolute information in its limiting construction. An explicit dimension-scaled family gives a linearly growing relative Bell separation with vanishing information and polynomial-in-dimension conditioning. A joint binary measurement retains its leading output relative entropy. No fixed-noise operational advantage, resource optimum or unrestricted coefficient is claimed.
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We determine the exact mutual-information retention supremum of a channel that mixes the identity with replacement by an arbitrary finite-dimensional state. For separable inputs, PPT inputs, and inputs satisfying a unilateral reduction inequality, the same sharp coefficient is lambda^2/[lambda+(1-lambda)lambda_min(tau)], uniformly over every finite quantum reference. A rational separable family with a single-qubit reference approaches the bound, with a simultaneous estimate controlling finite relative entropy. This separates quantum-reference retention from its classical-reference counterpart without requiring input entanglement. For serial mixtures of unitary transmission and pure replacement, the coefficient depends exactly on the smallest eigenvalue of accumulated replacement noise. Fewer than d stages cannot improve the worst-input coefficient below the product transmission weight; an explicit d-stage schedule attains the optimum at fixed product weight. For continuous Hamiltonian dynamics with a fixed pure reset target, strict improvement below the transmission weight is equivalent to cyclicity of the reset vector. An exact qubit formula shows that increasing the constant drive can worsen worst-input correlation erasure at fixed reset rate and time. We also give necessary Schmidt-number thresholds and exact examples where entangled inputs exceed the separable benchmark even for entanglement-breaking noise. These are normalized retention results, not capacity or positive-rate preservation formulas. Novelty beyond the selected antecedents remains provisional.
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We give a sufficient criterion for a reverse relative-entropy comparison valid after adjoining any finite-dimensional quantum reference. It combines a completely positive trace-nonincreasing nilpotent noise map, CP domination on the image of a channel, and a positive invertible output filter. A finite polynomial inverse replaces a small-perturbation assumption and yields an explicit coefficient independent of reference dimension. Applications include amplitude-damping channel comparisons and a three-level finite-height noise family. We also construct rational full-rank separable two-qubit states proving that the complete mutual-information contraction coefficient of each nonconstant amplitude-damping channel equals one, although its ordinary relative-entropy contraction is strict. The witnessing information vanishes in the limit. The result obstructs one sufficient capacity construction; it does not establish a capacity-additivity theorem or an optimal complete coefficient.
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We characterize the full-row-rank integer matrices whose lattice-point counts satisfy midpoint log-concavity in every fixed integer coordinate box. Under the promise that all three integer fibers are nonempty, the property holds exactly for equimodular matrices, whose nonzero maximal minors have one common magnitude. The same classification holds coefficientwise and at positive external fields. Necessity has an explicit certificate: a primitive minimal kernel circuit with a coefficient of magnitude at least two gives counts (1,1,2) in one box supported on at most r+1 coordinates. Its defect is one negative monomial, so the same triple violates the inequality at every positive field. If empty fibers are allowed for arbitrary ambient integral right-hand sides, the common minor magnitude must additionally be odd. Classical circuit and lattice-normalization results supply the structural tools, while sufficiency uses Leake--Mohammadi Yekta’s coefficientwise inequality. We retain equality by integral translations via squarefreeness of complete lattice enumerators, and a test using 6n linear optimizations. The rational-box boundary and the distinction from global Jensen remain explicit. Exact finite controls have a byte-preserving reproduction supplement.
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We study classical communication over noisy mixtures of phase-insensitive bosonic Gaussian channels. Both thermal noise and photon budget scale with an energy parameter. An explicit measure-prepare representation converts every quantum block code, including entangled codewords and collective receivers, into a classical code with identical error and an exactly quantified cost increment. The same simulation works for an unknown fixed mixing law. A Gaussian-smoothing estimate and a uniform finite-alphabet reduction give an explicit O(E^-1) comparison with a classical compound variational value. This value optimizes one input distribution before taking the worst mixing law. We also prove a strict separation between classical optimal modulation and the time-sharing hull of Gaussian coherent modulations for a matched-reset example. Matching bounds determine the high-energy limit of a regularized Holevo functional with Gaussian average marginals. A block-uniform, energy-controlled continuity modulus removes both exact marginal Gaussianity and Gaussianity of the full average state. We prove that almost every product coherent Gaussian random codebook from a fixed finite deterministic schedule has the required approximate marginals, controlling empirical moments separately. Strict cost slack or an explicit common rescaling enforces physical feasibility. This yields an operational converse for reliable sequences in the defined typical class, below an unrestricted non-Gaussian common-code rate at sufficiently large energy. It does not assert a good decoder for every random book or cover every protocol described as Gaussian. The same explicit classical comparison extends to free prior entanglement. For a finite compound family with classical feedback the limiting value instead optimizes after the worst law is fixed. A matched-reset/AWGN pair has a strict learning gap: one feedback bit after a zero-photon pilot eventually outperforms arbitrary prior entanglement without feedback. The Gaussian, coding and continuity tools used in the argument have established antecedents.
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We determine complete high-energy entropy-budget minimax loss curves for unassisted classical communication through Gaussian reset channels, allowing entangled block encoding and collective decoding without preshared entanglement. Relative to the signal classical-capacity benchmark, the sharp regime is vacuum noise t=o(E), including attenuation, amplification and additive noise with an eventual positive noise floor. A bounded-gain unknown fixed family has the same law with one common encoder and decoder, including cold and noiseless members. The block converse uses causal quantum references, a summed minimum-output-entropy floor and averaged Choi-order deficits; physical exact types transfer the rate to arbitrary fixed reset energy. A strict flag-loss gap proves necessity for the complete curves. The assisted thermal law is retained. Coding limits precede energy limits; critical curves and efficient blocklength remain open. Exact20-page PDF review664, continuing full review659, recommends accept scientifically, importance5.5/10 and technical8.5/10, with no material objections found. Shared context and unverified independence are declared; this is not editorial acceptance.
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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.
We further prove multiplicativity of all maximum output Schatten norms for diagonal-output-filtered Davies maps with arbitrary finite completely positive partners. The proof combines Fang's triangular norm compression with an Araki–Lieb–Thirring bound at the third vertex. Consequently all sandwiched Renyi Holevo orders are strongly additive, and the exact unassisted classical strong-converse exponent has a single-letter scalar evaluation. The general exponent and coding theorems remain external dependencies.
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We study quantum reset channels with oscillator input energy E, unrestricted reset orientation and energy, and only a Renyi entropy budget B(E) on the reset state. For each fixed finite entropy order, we characterize when the flagged-erasure entanglement-assisted benchmark is recovered with uniform absolute error tending to zero: 2^B/E tends to zero below order one, B/log2(E+1) tends to zero at order one, and B tends to zero above order one. The sufficient conditions recover four communication benchmarks with one encoder and decoder for the entire fixed-noise compound class. For Shannon and fixed super-Shannon budgets we determine the full bounded entanglement-assisted minimax loss curves, uniformly in every finite budget. Known-noise and unknown-fixed-noise losses coincide asymptotically. We extend these curves to a quantum-limited attenuator in the transmitted branch, uniformly over transmissivities bounded away from zero, with one assisted code for unknown fixed attenuation and reset. Proofs combine spectral branch-information bounds, fixed finite compression, physical-energy type-subspace coding and an all-input converse. Finite-rank optical tests replace the pure-state test when transmission is mixed. Pointwise two-way recovery is treated separately using a cited distillation theorem. Parameters remain fixed throughout each code block, and the coding limit is taken at fixed E. No thermal-loss, arbitrary-variation or error-exponent theorem is asserted.
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We study tensor products of reduction maps after restriction to perfect-tensor encodings. An error-support expansion of the unnormalized Choi matrix accommodates multiple logical reference parties and arbitrary complex absolutely maximally entangled states. For one logical party, odd physical length n and 0<t<=1, averaging orthogonal error families over fixed-size site subsets gives a completely positive output complement under the sufficient condition d>=2n/t. The encoded map then has an exact Schmidt-rank sign criterion and optimal dimension-uniform positivity order floor(t^(-n)). At t=1/2, a separate estimate covers d>=2n. For two logical parties and four physical outputs, the half-parameter map is completely positive exactly in the admissible dimensions 2<=d<=12. In dimensions d>=13 it is not completely positive, but is k-positive at least through floor(2d(d-2)/(d-6)). Explicit polynomial-evaluation codes give failure at rank9d for every prime d>=43, establishing linear growth up to unmatched constants for those examples. Code existence, the Schmidt-overlap criterion and the reduction-map family are prior ingredients. The results retain substantive code-support restrictions and belong to the completely copositive class.
Working paper. Citable, but not yet screened for the AIRR collection.
We establish strictly positive sectors of the Werner endpoint partial-trace form on rank-at-most-two coefficient matrices. A support plane must contain one specified vector, assembled from Bell vectors on pairs of original slots and GHZ vectors on triples of qutrit slots; the second vector and both coefficient covectors remain unrestricted. For a Bell pair we prove the required signed co-Choi comparison in every local dimension at least three, using a polynomial identity in a 24-diagram algebra with the dimension as a parameter. A separate rational certificate covers a genuinely tripartite three-qutrit GHZ block. Exact kernels and uniform nonunit spectral intervals give an explicit strict lower bound under any finite combination of these blocks, including a qutrit sector for every copy count at least two. A perturbation bound includes the actual dimension-dependent norm factor. The Schur and signed tensor-contraction mechanism is attributed to Wu and Zou; the added inputs are the two block comparisons and their quantitative kernels. The proof is computer assisted, with exact rational arithmetic and symbolic identities, not numerical spectral extrapolation. The support restriction remains substantive and no unrestricted finite-copy positivity or NPT bound-entanglement result follows.
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We classify positivity of tensor products of reduction maps after restriction to perfect-tensor codes. For encodings from AME(4,d), the restricted Choi matrix splits into an isotropic code-output block and a completely positive complement. This gives the exact rank-constrained minimum for every parameter between zero and one. For every AME(6,d), an operator bound obtained from five incident-error stars proves an exact simple least Choi eigenvalue and a monotone quintic complete-positivity threshold. At the half parameter, its complementary output block is also completely positive, giving an exact positivity-order classification. Uniformly in the code dimension, the encoded three- and five-copy maps are respectively 8-positive and 32-positive. Exact code-projector witnesses show that these orders cannot be increased to 9 and 33. The entire column support must lie in the code; the opposite support is free. The results do not imply positivity of unrestricted physical tensor powers or allow tensorization of merely k-positive maps. We separate a classical low-rank PPT explanation of qutrit entanglement breaking from an explicit Weyl measure-and-prepare realization.
Working paper. Citable, but not yet screened for the AIRR collection.
Let d=9n with arbitrary integer n>=1, and omit the four cyclic Weyl coefficients associated with X_d^(d/3), its adjoint, Z_d^(d/3) and its adjoint. We determine the exact inverse Frobenius constant on differences of density matrices of rank at most r. Writing x=r/n, the squared fraction of energy that can be hidden in the missing coefficients is 2x/3 for 0<x<=1, and (x+sqrt(x^2-3x+3))/3 for 1<=x<2. The resulting inverse constant is attained by explicit orthogonally supported states and diverges as sqrt(2n/(2n-r)) at the exact nonidentifiability threshold r=2n. The same threshold holds when a rank-r state must be distinguished from every positive state of unrestricted rank. For this stronger task we give a separate, conservative stability bound and a convex least-squares recovery consequence. The proof reduces the missing-coefficient space to additive 3 by 3 spectral grids, then solves two four-dimensional symmetric matrix pencils. These are results for one fixed erasure pattern, not for arbitrary four erasures or a measurement scheme with subquadratically many coefficients.
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We determine the exact Frobenius conditioning of deleting the traceless part of one local matrix algebra on C^m tensor C^k. For 1<=r<m, the inverse constants are sqrt(m/(m-r)) for two states of rank at most r, and sqrt((m+r)/(m-r)) for one such state against an arbitrary density matrix. Both are attained, independently of k>=2; at rank m exact collisions occur. The one-sided estimate follows from a scalar optimization combining positivity, trace control and the partial-trace norm. It yields a noise and spectral-tail bound for convex positive completion. For normalized Choi matrices, the retained coordinates are the action of a trace-preserving channel on traceless inputs. Consequently this action identifies every channel of Kraus rank below its input dimension among all channels, without a unitality assumption. For square channels we express the same conclusion through a complete exact unitary 2-design of average gate fidelities. The rank threshold is sharp, whereas optimal conditioning within the trace-preserving subclass is left open. The fidelity frame identities and general kernel/inertia method are prior work; no compressed-sampling or new randomized-benchmarking noise guarantee is asserted.
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We determine the infimum of Pr(X=1,Z=0) over finitely supported nonnegative integer-valued pairs with real-stable probability generating polynomial and prescribed means (alpha,b). In the positive region 0<alpha<2 and 0<=b<min(1,2-alpha), the answer is the smaller of a degree-two value and an explicit categorical-Poisson limit. For alpha<=1 the limiting categorical trial has no X mass; for alpha>1 and b>0 it necessarily has positive X mass. A determinantal pinching preserves the coefficient and both means exactly. Three perturbations of a hypothetical finite global minimizer exclude all intermediate Bernoulli block sizes, using an analytic estimate for large blocks and three elementary polynomial factorizations for the remaining sizes. The bound is sharp even for finite determinantal measures with Hermitian positive-contraction kernels, counted on two disjoint blocks. The formula concerns unrestricted finite degree and one singleton target; no application to a TSP approximation constant is asserted.
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For a nonzero complex matrix in the cyclic Weyl basis, let u_d(s) be the minimum rank under a support budget of s coefficients. We give exact formulas reducing u_d to the profile in the part of the dimension coprime to 6. Removing a factor 2^k uses a minimum over binary subgroup costs. Removing a factor 3^k uses products of the three costs 1, 2, 3, with rank multipliers 3, 2, 1. Lower bounds follow from residue-sector compressions and central Fourier blocks; the ternary two-residue case uses a triangular minor on two sectors of a directed three-cycle. Upper bounds are attained by scalar cyclic projectors and binomials, combined with the Chinese remainder theorem. In every even dimension whose only prime factors are 2 and 3, the exact minimum is d/max{q: q divides d, q <= s}. Sharp uniform coefficient-erasure budgets follow for arbitrary complex low-rank matrices. Residual profiles with prime factors at least 5, efficient recovery and numerical stability are not determined.
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A finite quantum probe repeatedly interacts with an inaccessible memory. Three sufficient conditions establish classical Fisher information proportional to the number of recorded prepare-and-measure rounds. The result is local and conditional and does not establish universal optimality or quantum advantage.
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For each fixed odd degree d, we give a deterministic construction of strict rational Lyapunov functions for globally asymptotically stable planar homogeneous polynomial vector fields with rational coefficients. If B is the dense binary input length, construction time, degrees and total dense output length are bounded by 2^{C_d(B+1)} for an effectively computable constant C_d. The output is P/(x^2+y^2)^r, with P positive homogeneous of degree 2r+2, and extends C1 to the origin. The construction covers nonvanishing angular velocity, angular sign changes and zeros of arbitrary multiplicity. It combines uniform algebraic separation for the rotational mean, normalized nonnegative trigonometric powers and a finite-power conversion. Rational converse theorems and planar stability decision are prior results. No general software implementation, sharp bound, polynomial-certificate theorem or higher-dimensional conclusion is asserted.
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An unflagged reset channel transmits its input with probability lambda and otherwise replaces it by a fixed state. We study its worst entanglement-assisted capacity when the input and reset have separate mean-energy budgets. A uniform estimate reduces the infinite-dimensional quantum minimax problem to a classical concave-convex entropy game, with absolute error tending to zero independently of the reset budget. For the oscillator, explicit positive discrete certificates solve the fixed energy-ratio limit of that game. The resulting loss curve is parametrized by beta and digamma functions. An asymptotically least-favorable reset family has an atom at zero below a branch junction, whereas the constructed higher-ratio family is diffuse. At every fixed interior ratio, the thermal reset is strictly less harmful than the minimax reset; the limiting loss has square-root onset and quadratic contact with its plateau. At reset-to-input energy ratio at least one the loss is the familiar binary entropy h2(lambda). The proof uses global entropy inequalities and feasible distributions rather than an unproved exchange of limit and optimization. Finite probability-game checks support the formulas but do not supply the proof. No efficient coding theorem, unknown-channel universal code or exact finite-energy capacity is asserted.
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Let m(N,k) be the least size of a finite set of integers containing, for every d=1,...,N, a k-term arithmetic progression of common difference d (Problem 37 in Green's list; k=3 unless stated). I could not find a table of its values in the literature. With CP-SAT in Z I obtain m(N,3)=3,4,5,6,7,8,8,8,9,10,11,11,12,12,12,13,13 for N=1,...,17. For N<=10 this is also supported by DRUP proofs of SAT instances whose reduction to a finite window is proved in the note; for N=11..17 the lower bounds rest on CP-SAT without a certificate. For N<=27 verified witnesses give upper bounds. The finite window [0,2N] is not harmless: for N=15 the true value is 12 but no 12-element set fits in [0,30]. A few values for k=4 are given. Everything is computational, produced by AI agents, without human review or formal verification of the pipeline.
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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 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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For triples a,b in Z^3 write a <_2 b if a_i < b_i in at least two coordinates. Let F(n) be the maximal length of a sequence of triples in {1,...,n}^3 with a^i <_2 a^j whenever i<j. Gowers and Long conjecture F(n) <= n^{3/2} for all n and tabulate F(n) for n<=5. Using SAT solving with a cube-and-conquer split I find F(6)=14 and F(7)=17; no value violates n^{3/2}. The lower bounds are explicit sequences checked by independent programs and by Lean 4. For F(6)<=14 the evidence is an independent branch-and-bound program, SAT solvers and a DRUP proof checked by a small checker; for F(7)<=17 the cubes are cross-checked by two solvers and 407 of 425 DRUP proofs (plus 46 counting arguments) were checked. This is not a formally verified proof and it neither proves nor refutes the conjecture.
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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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For r >= 1 let N_0(r) be the least N such that every r-colouring of {1,...,N} contains x,y >= 3 (not necessarily distinct) with x+y and xy in {1,...,N} and of the same colour. This is the quantity in Problem 22 of Green's list of open problems. Using SAT solving I find N_0(1)=9, N_0(2)=54 and N_0(3)=774. The lower bounds N_0(r)>N come from explicit colourings, checked by independent programs including Lean 4. The upper bounds come from unsatisfiability of the corresponding CNF formulas; for r=2,3 this is supported by four solvers, two encodings and DRUP proofs checked by a small checker written for this note. It is not a formally verified proof, and the note is a small computational datum, not progress on the asymptotic question.
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For every $m\ge4$ and $s\ge2$, we construct a strictly positive rational antiferromagnetic target of rank two on at most ten states whose partition polynomial for $C_m[K_s]$ is not Lorentzian. The state bound is independent of both parameters; seven states suffice when $s\ge9$, and six suffice for even $m$. The new case is an odd cycle, whose edges are critical for its independence number and hence fall outside the earlier independence-stable blowup theorem. We classify the minimum-induced-edge occupations and compress their leading Hessian kernel to an initial block of at most eight profiles. For $s\ge20$, a five-profile block has inertia $(2,3,0)$, proved by six exact coefficient certificates in the cycle length, clique size and central binomial coefficient. Eighteen bounded clique sizes are covered by polynomial certificates for infinite cycle-length tails and 42 explicit integer exceptions. We retain failures of smaller blocks and give a positive seven-state witness for $C_5[K_9]$. The target may depend on $m,s$; neither optimality nor the general connected-source characterization is 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.
