For a traceless Hermitian matrix F, this paper minimizes the product of the unnormalized Hilbert--Schmidt norms of Hermitian H and K satisfying -i[H,K]=F. On the complete one-spike spectral cone with nonzero spectrum (P,-b_1,...,-b_n), it proves the exact formula kappa_d(F)=sum_j j b_j, fixes the nonzero singular spectrum and rank of every balanced optimum, derives sharp trace-distance stability and strict Schur concavity, and remains invariant under ambient zero padding. For every finite-dimensional traceless Hermitian target, it also proves exact balanced-loop laws A_3(F)=12 sqrt(3) kappa_d(F) and A_4(F)=16 kappa_d(F), giving a universal 23.02 percent fourth-kick reduction. The work does not give a closed formula when both sign multiplicities exceed one, classify all optimizing matrices, or claim exhaustive bibliographic priority.
For a nonzero traceless Hermitian matrix F, this paper studies the least product of unnormalized Hilbert--Schmidt norms of Hermitian H and K satisfying -i[H,K]=F. Writing rho=rank(F) and P(F)=||F||_1/2, it proves the sharp dimension-free bounds P(F)<=kappa_d(F)<=rho P(F)/2. Both equality loci are classified: the lower endpoint is attained exactly by centrally paired nonzero spectra, while the upper endpoint is attained exactly, up to positive scale and sign, by the one-spike spectrum (rho-1,-1,...,-1), with arbitrary zero padding. An exact sign-cut leakage identity identifies the full tax above the trace-norm floor. The upper bound uses an exact weighted-shift permutation average, and rho-1 explicit Horn--Littlewood--Richardson inequalities certify upper-endpoint sharpness. The work does not claim a closed formula for general interior spectra or an upper-endpoint stability modulus.
A sigmoid neuron induces two distinct spectral geometries: a population radial-tangential anisotropy controlled locally by the Schwarzian derivative, and an empirical extreme-saturation hierarchy controlled by samples nearest the transition hyperplane. For logistic powers h_p=sigma'^p, p>0, we prove exact saturation constants, sharp pointwise bilateral oracle-Gram complexity, and regularly varying spherical radius phases. We then make the empirical hierarchy quantitative. A deterministic exterior-power theorem resolves every leading eigenspace; an exact inverse-Gaussian angle law and a finite hierarchy perturbation lemma yield an explicit joint finite-radius, finite-sample guarantee for the bottom eigenspace and teacher direction. Finally, a two-dimensional tangent Gaussian process proves that uniform control over a full angular shell intrinsically incurs a square-root log-radius factor in the iterated high-probability limit. Deterministic replays, hostile proof, novelty, and reproducibility audits, and exact source provenance accompany the paper.
Given d >= 1, 1 <= s <= m, N = binom(d+m,d), and any N isolated complex hypersurface germs g_i of order at least s+1, we construct, for every D >= 1 + sum_i(tau(g_i)+2), a smooth integral degree-D hypersurface with a tangent hyperplane whose reduced Gauss fibre has exactly N points, is point-span order-s osculating-absorbing, and has the prescribed completed Tjurina algebras. The fibre length is sum_i tau(g_i), and the dual multiplicity is sum_i mu(g_i). For surfaces, the classical complete Tjurina spectrum of ordinary plane multiple points yields every integer fibre length between binom(m+2,2) floor((3s^2+4s-3)/4) and binom(m+2,2)s^2 at fixed dual multiplicity. We also give a direct strong-Lefschetz proof that x^(s+1)+y^(s+1)+z^(s+1)+(x+y+z)^(s+2) attains Wahl's classical three-variable value s(s+2)(2s-1)/3, producing sharp absorbed threefold fibres. All claims are over C; degree bounds are sufficient, not asserted minimal, and no exhaustive priority or human peer-review claim is made.
Let h in C[[x_1,...,x_d]] have order at least s+1 and finite Tjurina algebra. Euler cancellation replaces h, without changing its Tjurina ideal, by a generator of order at least s+2. A finite-jet generator count then gives tau(h) at least E_{d,s}, the maximum over K of binom(d+K,d)-d binom(d+K-s,d)-binom(d+K-s-2,d). For plane germs this is the exact universal floor floor((3s^2+4s-3)/4), attained by classical ordinary multiple-point families. Applied to a point-span s-osculating-absorbing reduced fibre of the Gauss map in the complete O_X(m) embedding, the result gives length(Gamma_eta) at least E_{d,s} binom(d+m,d). For surfaces, a prescribed-jet construction realizes equality at an explicit sufficient degree. The work is over C; the higher-dimensional floor need not be sharp, the degree threshold is not claimed minimal, and no exhaustive priority, human peer-review, or formal-verification claim is made.
Let X be a smooth non-linear complex hypersurface and let Gamma_eta be the scheme-theoretic fibre of its Gauss normalization over a tangent hyperplane. The completed local fibre algebra is the Tjurina algebra of the tangent-section germ, while the classical dual multiplicity is the sum of its Milnor numbers; hence their difference is the total Tjurina–Milnor defect. Under point-span order-s osculating absorption in the complete O_X(m) embedding, the fibre contains the order-s fat point at every reduced support and has length at least binom(d+s-1,d) binom(d+m,d). When d(s-1)>s+1, prescribed jets realize every integral defect from 0 through the extremal support size while keeping all Milnor numbers fixed. The degree threshold is sufficient, not minimal; no positive-characteristic extension, exhaustive priority claim, human peer review, or formal verification is claimed.
Let X be a smooth complex hypersurface and let Z be the complete reduced fibre of its Gauss map over a tangent hyperplane W. In the complete O_X(m) embedding, assume that the point span of Z contains the order-s osculating space at every support, with 1 <= s <= m. We prove the sharp floor mult_W(X^vee) >= s^d |Z| >= s^d binom(d+m,d). Absorption forces order-(s+1) contact; the classical multiplicity-Milnor formula and a local Milnor lower bound give the dual multiplicity estimate, while exact tangent absorption gives the branch floor. For every d,m,s we construct equality examples with proper point span, exact reduced Gauss fibre, ordinary section singularities, and the stated weighted tangent-cone cycle. The construction gives a sufficient nonminimal hypersurface degree. No equality classification, positive-characteristic extension, exhaustive priority claim, human peer review, or formal verification is claimed.
Let X be a smooth projective integral d-fold over an algebraically closed field, let H be very ample, and use the complete embedding defined by H^m. Fix 1 <= s <= m and suppose that the span of a nonempty finite reduced set Z contains the order-s affine osculating space at every support. We prove the exact characteristic-free floor dim(S_Z), |Z| >= binom(d+m,d), by reduction to exact tangent absorption. For every d, m, s and every characteristic, we construct a smooth integral hypersurface with a proper-span equality set whose normal coordinate vanishes to order s+1 at every support. If r_1(Z) is the degree-one evaluation rank and m >= 2s+1, we also prove the rank-sensitive term binom(d+s,d) r_1(Z); rational normal curves show the threshold is necessary for that uniform formula. The earlier mixed-jet certificate is retained for jet-ample polarizations but identified as numerically subordinate for tensor powers. No equality classification, minimal hypersurface degree, exhaustive priority claim, human peer review, or formal verification is claimed.
Let X be a smooth projective integral d-fold over an algebraically closed field of arbitrary characteristic, let L be very ample, let m be at least one, and let Z be a nonempty finite reduced set whose point span in the complete L^m-embedding contains every tangent space at its supports. We prove the exact codimension-free floor h_Z(m), |Z| >= binom(d+m,d). An adapted finite projection to P^d transfers absorption to equality between the degree-m pieces of the radical ideal of the projected points and its symbolic square. The standard perfect-field derivative/p-th-root gap forces that piece to vanish. Simplex-lattice interpolation and characteristic-zero Bertini yield proper-span equality examples over C in every dimension and degree. No imperfect-field, singular, nonreduced, or equality-classification result is claimed.
Let X be a smooth projective integral d-fold over an algebraically closed field and use the complete embedding defined by H^m, m at least 3. For a nonempty finite reduced point-span tangent-absorbing set Z, this paper proves in arbitrary characteristic the lower bound dim(S_Z), |Z| at least max{J(d,m), min{(d+1)(d+2), binom(d+m,d)}}. The positive-characteristic Gauss-branch argument replaces the characteristic-zero derivative contradiction by Frobenius and radicality. The paper also proves injectivity of the original Gauss map for suitable complete factorized polarizations and, over the complex numbers, constructs smooth hypersurfaces attaining the binomial estimate inside the exceptional Gauss branch for every d at least 2 and m at least 3. The construction is existential; no positive-characteristic Bertini realization, equality classification, human peer review, or exhaustive priority claim is made. A related ARR paper proves a sharper exact global binomial floor in characteristic zero.
A quantum list measurement succeeds when its output contains the prepared label. Building on the known equivalence between learning width and Gram-matrix factor width, this paper closes an exact realization-sensitive branch. Every full-spark ensemble of N pure-state rays spanning C^r that admits strictly positive tight representatives has minimum zero-error list size N-r+1. Weighted Hodge duals of all (r-1)-fold frame wedges give an explicit attaining POVM, and a physical-space annihilator-cone formulation gives constructive compression to at most r^2 outcomes. A smallest-eigenvalue functional supplies a positive and perturbatively stable Bayes-error floor below threshold. For a flat consecutive-support Schmidt-rank-r probe of the complete d-dimensional Weyl-channel ensemble, the fixed-probe threshold is d-r+1. When r divides d, an arithmetic-support construction and dimension converse close the optimization over all pure Schmidt-rank-r probes at d/r. For the consecutive rank-two probe, the complete one-shot Bayes list curve is determined exactly. The nondivisible probe optimum, adaptive or multiuse testers, asymptotic capacity, arbitrary mixed states, and non-scalable full-spark ensembles remain outside scope.
Let G be a compact connected Lie group with a bi-invariant metric. For the two-dimensional heat-kernel Yang--Mills model, this paper gives an explicit finite chain from a boundary-conditioned edge integral on an arbitrary regular annular cellulation to the gauge-invariant class-function transfer kernel. Conditional Haar coordinates make the boundary constraints and boundary-edge subdivision precise. Edge and face subdivision invariance, an explicit PL radial-cut construction, and a primal-tree/dual-cotree disk-elimination schedule yield a cellulation-independent orbital heat-kernel integral. Peter--Weyl theory then diagonalizes the transfer operator with multipliers exp(-t c_lambda). The exact mean-zero norm is exp(-t c_*(G)), the gap of I-T_t is 1-exp(-t c_*(G)), and the equality sector consists of all irreducible characters with minimum positive Casimir. In the stated SU(2) normalization the sharp exponent is 3/4, while the induced SO(3) quotient removes the fundamental channel and raises it to 2. The work assembles and audits classical two-dimensional Yang--Mills ingredients; it does not claim a new character solution, four-dimensional mass gap, continuum reconstruction, or clustering for arbitrary bulk observables.
In quantum list discrimination a measurement returns at most ell candidate labels and succeeds when the true hypothesis belongs to the returned list. We associate a Rado independent-transversal matroid to the support subspaces of a mixed quantum ensemble and prove that every true-label inclusion vector lies in the independence polytope of the ell-fold matroid union. This yields all-subset Bayesian bounds for arbitrary priors, soft rewards, an integer congestion deficit, equality audits, and canonical compression to quantum-process testers. Exact attainment and insufficiency examples separate support combinatorics from quantum geometry. We then solve two input-dependent process families. For binary laminar dephase-prepare channels with M=2^h, list size ell=2^s, and q uses, arbitrary entangled parallel probes and adaptive quantum memories obey P_parallel=min(1,ell(q+1)/M) and P_adaptive=min(1,ell 2^q/M). For complete unitary-error ensembles we translate the known approximate dense-coding spectrum law into a list cap, derive an exact serial/parallel/Bell multitime trichotomy, and give a fixed-probe example where the coarse list-rank cap is not attained. The matroid theorem is a support obstruction rather than a general POVM feasibility characterization; the laminar separation is a classical feedback tradeoff embedded quantumly, and no indefinite-causal-order advantage is claimed.
Let P be a Haar-distributed complex Grassmann projector and consider the matrix--Bingham normalizer Z_A(s)=E exp{s tr(AP)} on the trace-zero, Frobenius-unit external sphere. We prove exact complementary results for its canonical two-level branch geometry. On every half Grassmannian Gr_C(r,2r), the balanced rank-r field strictly maximizes the normalized two-block moments of degrees 4, 6, 8, and 10. Exact endpoint analysis gives balanced dominance at small and large field, while a finite certification theorem reduces coefficientwise all-field order at each fixed rank to a terminating block of rational Hankel signs plus an analytic tail. At Gr_C(3,6), a closed all-degree argument proves strict balanced order for every field. At Gr_C(4,8), exact rational arithmetic through degree 268 and the analytic tail give a second all-field theorem. On Gr_C(2,5), compact overlap densities, Sturm arithmetic over Q(sqrt(6)), and strict total positivity prove a unique simple exchange in the complete oriented two-level family. The results do not classify arbitrary multi-level external spectra and do not claim an unrestricted all-distortion rate--distortion function.
Let a compact group act orthogonally on a sphere and let mu be the invariant law of a proper antipodal orbit. We study the exponential orbital potential. At a point of the source orbit, every spherical-harmonic coefficient of the orbit average is a squared norm, giving an exact negative spherical-Laplacian identity at every nonzero field. Provided the source orbit is critical--automatically so when the normal isotropy representation has no fixed vector--irreducibility, or a transitive symmetry of its irreducible normal blocks, upgrades this trace identity to strict Morse--Bott maximality for every field. An explicit torus-orbit counterexample shows why criticality cannot be omitted. Applied to centered projector embeddings of the real, complex, and quaternionic half-Grassmannians, the theorem proves all-field local rigidity of the balanced matrix--Bingham branch and excludes a radial spinodal. In the complex case, exact Jacobi-generator identities yield constrained-Hessian operators for every two-block external spectrum and a sharp weak-field metastability threshold at k=2r/3. A Stein inequality closes every sufficiently high Taylor degree at each fixed multiplicity. The results constrain unresolved intermediate phases but do not claim global spectral optimality, a complete finite-field phase diagram, or an all-distortion rate--distortion function.
We give a finite-sample test for whether one joint Gaussian sketch covariance is compatible with a positive-semidefinite matrix-valued measure supported on a prescribed interval. The observed vector contains several exponent blocks, so its population covariance is one block-Hankel Gram matrix rather than a list of unrelated moment estimates. The classical truncated matricial Hausdorff theorem supplies the exact parity-dependent null. A single Gaussian singular-value event gives a simultaneous Loewner band for the whole covariance; intersecting that band with the structured moment cone yields a nonasymptotic level-alpha semidefinite test without sample splitting. The full joint Gram is strictly more informative than the integer-time localizer used in the preceding finite-sample method: an explicit two-atom family satisfies the old population condition but violates the half-time condition. We prove opposing finite-sample power guarantees above an explicit threshold on the same acquisition. At regular boundary points, the constrained likelihood ratio converges to squared Gaussian distance from an explicit spectrahedral tangent cone, giving pointwise local power and a matching root-n separation boundary. We also give auditable dual semantics and an exact rational certificate for the strict fixture. Classical moment, concentration, and constrained-likelihood ingredients are attributed explicitly; the contribution is their structured joint-sketch integration and strict same-data separation.
Minimum-error discrimination of quantum processes is normally optimized over testers whose normalization may encode parallel, sequential, or indefinite-order access. For a fixed physical deterministic normalization, Moore-Penrose compression maps the tester exactly to a POVM on normalized effective states and preserves every conditional probability. We associate to the support subspaces of arbitrary positive process operators a Rado matroid on the hypothesis labels and prove that every correct-label probability vector lies in its independence polytope. Consequently the Bayes success probability is at most the prior weight of a maximum-weight independent transversal. A robust extension replaces exact supports by arbitrary positive low-rank cores and charges only the prior-weighted worst-case discarded tester mass; valid full-rank process admixture of weight eta degrades the certificate by at most eta. We give an explicit reduction to linear matroid intersection, an equality audit at strict prior drops, a deterministic Gram criterion for perfect rank-one discrimination, and exactly solved qubit phase-gate families. In a five-channel instance the exact general-tester optimum is 0.80 while the total-dimension relaxation is 0.90. The result is a support-based upper bound and is not claimed to determine every mixed-process optimum.
Fix a nonzero dominant weight lambda and let the highest weight grow along the ray N lambda. We study the exact classical Shannon rate-distortion function of the invariant phase-lifted coherent orbit in V_{N lambda} under squared ambient Hilbert-space loss. The finite-N scalar formula is known; the new problem is the coupled large-weight and distortion limit. If d_N is the dimension of V_{N lambda}, ell_N=log d_N, and a=dim_C(G_C/P_lambda)+1/2, we prove that the unique global origin-tangent contact, for all sufficiently large N, obeys t_c=2 ell_N+2a log ell_N+log(4 pi)-a+o(1), while the contact distortion is a/ell_N+o(ell_N^{-1}) and its time-sharing slope is ell_N+a log ell_N+log(2 sqrt(pi))+o(1). All root-system and Weyl leading constants cancel in these dimension variables. For fixed 0<D<=1, the exact unrestricted RDF satisfies R_N(D)/ell_N -> 1-D. On the noncommuting boundary scale D=x/ell_N, the centered rate converges locally uniformly to an explicit two-branch profile, with a curved high-fidelity branch tangent to a linear coexistence face at x=a. We also identify the soft-activation window and an exact fixed-radius Legendre expansion. The result is classical rate-distortion for an embedded coherent-amplitude source, not quantum rate-distortion or a derivation of Born's rule.
The standard k-query dimension bound for an unknown d-dimensional unitary uses the entire degree-k symmetric tensor space and scales as k^(d^2-1). We refine that bound to the degree-k Hilbert function of the projective oracle variety and compute the resulting support exactly in two physically structured families. For a fixed-angle qubit rotation with unknown axis, the repeated-query span has dimension (k+1)^2 on the generic branch, binomial(k+2,2) on the traceless branch, and one on the central branch. A compact-orbit leverage argument inserts this exact support into the general-tester bound, applying to the parallel, sequential, and mathematically admissible indefinite-order strategies covered by the standard tester model. The associated spherical and planar-axis frames admit closed harmonic spectra, purity formulae, endpoint cascades, and a complete tightness classification. For d-level selective-phase oracles, the projective closure is a Segre variety and the exact support is binomial(k+d-1,d-1)^2, reducing the ambient exponent from d^2-1 to 2d-2. These results quantify geometry-dependent query support rather than physical memory or an achievable discrimination probability in every finite ensemble. The work excludes inverse-oracle access, controlled bypasses, noise, finite-sample estimation, and tester models beyond those explicitly stated.
Let V_lambda be a finite-dimensional irreducible unitary representation of a compact connected semisimple group, and let X be uniform on the phase-lifted orbit of a highest-weight vector. We determine the classical Shannon rate–distortion function under squared ambient loss for arbitrary standard-Borel descriptions and decoders unrestricted to the orbit. Sugita's known integer-moment coherent-state extremum, combined with positive phase–Bessel resummation, yields an all-field fixed-radius Laplace order and exact equality classification. The vector problem reduces to an exact scalar radial envelope. Covariant tilted channels attain exposed points and revealed binary flags attain nonexposed chords. A representation-dimensional criterion forces discontinuous directional-information onset, while Weyl dimensions give a sharp root-system high-fidelity constant. For exterior powers, the source is a phased fermionic Slater determinant: the normalizer is hypergeometric, squared Pluecker overlap is a beta product, and every interior family with n at least 6 has discontinuous first activation. A distinct projective corollary treats intrinsic Pluecker distortion with reproduction restricted to the coherent orbit. The results concern classical phase-sensitive amplitude reconstruction, not quantum rate–distortion, click-only Born statistics, or particle dynamics.
Four balanced Pauli kicks along the vertices of a regular tetrahedron, followed by their reverse echo, generate 24 order-labelled qubit unitary channels. The words share a trace and form two 12-element orbits of the proper tetrahedral group. One query sees only their common trace. At two queries, orbit-dependent quadratic energies yield closed formulas for the Bell-square and globally optimized parallel strategies. At the algebraic point q=1/2, every causally ordered two-query protocol has success at most 9/24 because the fixed trace removes one dimension from the universal quadratic query space. An explicit rational two-comb normalization, a six-Kraus realization, and an exact weighted-frame identity attain the cap, giving P_causal=3/8 and P_parallel=(5+sqrt(15))/24. A uniform Lipschitz estimate proves that the same causal tester remains strictly better on an explicit open pulse interval. The result is an analytic adaptive advantage for a concrete non-group ensemble. It does not address indefinite causal order, noise, finite statistics, or arbitrary qubit ensembles. The central causal certificate is replayed exactly in quotient-ring arithmetic; the supplied word/collision and representation scripts are mixed symbolic-numerical corroborating audits.
Let W = x wedge y be the signed unit Pluecker coordinate of a Haar oriented two-plane in R^n, and let reproduction range over the full posterior-mean body under squared ambient loss. An all-field extremum theorem reduces its unrestricted Shannon rate--distortion function to phi_lambda(b) = K_q(2 lambda b) - lambda b^2, where q = n-2 and K_q = log(q! sum_{j>=0} kappa^(2j)/(q+2j)!). We solve the remaining global phase problem in every dimension. For J_q = L_q L_q' - kappa L_q L_q'' + kappa(L_q')^2, its coefficient of kappa^(2r-1) is a positive factor times r - E[d^2], where d follows a central parity-truncated binomial law. Likelihood-ratio ordering and an exact tail estimate prove one coefficient sign change for every q >= 4. Consequently the stationary curve has exactly one fold, the zero and positive phases have exactly one coexistence contact, and no later exchange or radial reentrance is possible. The RDF is therefore an explicit linear face followed by one covariant parametric branch for every n >= 6; together with the continuous cases n = 3, 4, 5, this completes the all-dimensional phase diagram. The unique contact has an exact large-dimension limit and an explicit logarithmic finite-size displacement. We additionally prove that the same RDF is the minimum worst-source channel capacity, that arbitrary joint classical memory costs exactly N R_n(D) on blocks of length N, and that transitivity makes the tilted information constant, giving a zero-dispersion strong converse. The results concern signed Pluecker-coordinate compression, not quantum rate--distortion or Born-probability prediction.
At distinct boundary frequencies, let a square rational-inner multiport route one fixed input ray to prescribed output rays. We prove that the minimum McMillan degree equals the least degree of a base-point-free projective curve through the ordered target rays, independently of ambient port count. For orthogonal target bands we solve line-to-subspace incidence constraints by full-support Lagrange kernels and derive a closed generic codimension law. Exact memory is the maximum shifted band cost, whereas border memory is the minimum and may be arbitrarily smaller. An intrinsic incidence matrix gives a calibrated singular-value error floor, a determinantal zero-error closure, and an all-data base-point deletion law with an incremental exact certificate. Finally, every zero-error sequence below exact memory has divergent peak dimensionless Wigner-Smith delay; any finite delay cap restores compactness, a positive attained error, and a three-phase operational classification. Explicit planar strata and exact rational certificates audit the results.
Suzuki's localized Weil form is represented by a lower-bounded self-adjoint operator A_a on L2(-a,a); nonnegativity for every a > 0 is equivalent to the Riemann hypothesis, while positivity at one fixed support is an unconditional and strictly weaker problem. We give a source-level interval certificate proving A_0.72 >= 5.890 x 10^-17 I > 0. Nested-core monotonicity gives the same scalar lower bound for every 0 < a <= 0.72. The proof decomposes the exact prime-power translation graph through n = 4 into thirteen intervals and bounds its infinite complement by a mode-sensitive Schur estimate. It isolates degrees 12 through 23 before controlling [24,176) and [176,infinity); both parity Schur matrices have 78 certified positive directions and no unresolved direction at 512-bit Arb precision. We also prove an exact Gauss-Stieltjes hierarchy for the logarithmic boundary potential, complete for strict positivity at every fixed support, and a multiband Loewner majorant requiring only O_epsilon(log log M) bands through degree M. The revised manuscript prints a formula-level source-to-Gram specification. A separately written program importing neither project modules nor python-flint reconstructs the prime-power graph and parity maps and independently reassembles the exported Schur balls with positive Weyl margins in both sectors. This is a bounded-support theorem, not a proof of the Riemann hypothesis.
We solve a finite-dimensional external-spectrum optimization problem for the complex matrix–Bingham law and use it to determine an unrestricted Shannon rate–distortion function. If P is Haar on Gr_C(2,4) and A is traceless Hermitian with fixed Frobenius norm, then E exp[s tr(AP)] is, for every s>0, uniquely maximized, up to unitary conjugacy, by the balanced spectrum (R/2,R/2,-R/2,-R/2). The exceptional model Gr_C(2,4)=Gr_2^+(R^6) turns the orbital integral into a positive series whose coefficients share one simplex-vertex maximizer. For the Haar rank-two state source rho_P=P/2, conditional least squares places arbitrary reports in the full body {sigma: 0<=sigma<=I/2, tr sigma=1}. We obtain its exact classical rate–distortion function for every 0<=D<=1/4. Beyond a scalar dual representation, we prove the complete radial phase diagram: one fold, one positive coexistence contact, no reentrance, and an exact two-piece frontier consisting of one time-sharing segment and one matrix–Bingham branch. The proof reduces the fold derivative to a power series with exactly one negative coefficient followed by strictly positive coefficients. Covariant channels attain the frontier, and the same value is the source-universal worst-state capacity; arbitrary joint n-block memories cost exactly n times the one-letter frontier. The result concerns classical memory for calibrated Born-probability prediction, not quantum rate–distortion, click simulation, or a derivation of Born's rule.
Finite Euclidean correlation matrices are routinely converted into spectral gaps by generalized eigenvalue, Prony, or Lanczos procedures. We determine a finite-sample boundary for what such data can falsify and what they cannot certify by non-rejection. Block Hankel pencils give monotone lower bounds on the visible transfer edge, while exact rank stabilization recovers the complete visible spectrum. An atom of weight w at any x*=exp(-g) changes every moment by at most w while forcing gap at most g; arbitrarily small strictly positive gaps therefore remain hidden even when an extra zero mode is forbidden. For a fixed nonsingular Gaussian experiment we compute the exact hidden-atom likelihood and derive its LAN information, sharp local power envelope, and powerless/local/consistent phase at the w sqrt(n) scale. Under a disclosed visibility floor, a fourth-kind Chebyshev filter solves the weighted localizer minimax exactly; a two-atom measure attains the bound, making the uniform sign threshold necessary and sufficient in the declared polynomial class. This margin feeds an exact-level two-coordinate Wishart test and a closed depth–sample resource law. A distribution-free companion uses paired differences and the exact quadratic range to handle bounded iid readouts with unknown mean and same-sample selection over a finite filter bank. A dependence-robust extension treats one stationary bounded beta-mixing trajectory: sparse pairing, Berbee coupling, and an explicit lag-covariance correction give finite-sample level, power, and total chain-horizon bounds under an externally certified mixing envelope. An interacting ANNNI-chain experiment through 2^16 states demonstrates exact symmetry blindness and multichannel recovery. Rejection falsifies an overstated gap; non-rejection alone is not a positive gap certificate.
We give a single exact state-counting theory for finite passive spectral routing tables. Fix distinct boundary frequencies, one input k-plane, and a word whose symbol a requests a target k-plane Y_a exactly n_a times. If the used targets are in direct-sum position—they need not be orthogonal and may have arbitrarily small principal angles—then the minimum McMillan degree among square finite rational-inner interpolants is k(L - min_a n_a). Thus the rarest target fixes generic passive memory, independently of node spacing and word order. The lower bound uses block-dual detectors: direct-sum geometry supplies a constant compression that is invertible on one target and annihilates every other target. Its determinant has k forced zeros at every wrong node, while a Blaschke–Potapov minor has no more zeros than the network has states. The upper bound is a matrix spectral compiler. Target-weighted node polynomials form a full-rank polynomial column; matrix Fejér–Riesz factorization normalizes it to a rational-inner column of degree at most the lower bound, and a degree-preserving lossless completion closes the network. Beyond the direct-sum locus we prove a detector-rank hierarchy, a maximum weighted hyperplane-occupancy bound for lines, and the complete three-line phase diagram. We also give a span sandwich, open-dense genericity, fail-closed noisy certification, an exact collision discontinuity, and the optimal integrated Wigner–Smith delay. Producer and independent verifiers audit nonorthogonal scalar and block tables, collision openings, and singular-incidence fixtures.
We give a unified exact rate–distortion theory for retaining the complete rank-one Born-probability field of a Haar-random pure state in complex projective space. A constrained root count, a sharp centered power-sum theorem, and Newton's expansion prove an all-degree spectral result: at fixed traceless Frobenius norm the projective Laplace transform is maximized by a one-positive-spike spectrum. This reduces the unrestricted Shannon rate–distortion function to an exact scalar complex-Bingham envelope in every dimension, with covariant channels and independently flagged coexistence mixtures attaining every distortion. For dimension at least three, directional information turns on discontinuously; the qubit curve and the all-dimensional high-fidelity constant are explicit. We then solve the full fixed-normalized-distortion limit. If y_* > 2 solves y_* - 1 = 2 log y_*, the limiting free energy has a unique coexistence point, and the information cost per dimension is a closed two-piece function with a nontrivial linear face. The finite-dimensional onset multiplier equals alpha_* d minus [alpha_*/(2 log y_*)] log d up to an O(1) remainder. Finally, a pointwise Hilbert projection converts every measurable scalar reporter into a physical density-matrix reporter without increasing risk for any pure input. Hence the exact worst-state channel capacity equals the Haar rate–distortion function, arbitrary joint n-state memories cost exactly n times the one-state frontier, and transitivity forces zero rate dispersion with an exponential finite-blocklength strong converse. These results concern reusable calibrated probability fields, not click-only simulation, state update, or intrinsic randomness.
Let P be Haar on the complex Grassmannian Gr(r,d) and set rho_P=P/r. We solve two coupled finite-dimensional problems. First, on the sphere of traceless Hermitian external fields with fixed Frobenius norm, we determine incompatible weak- and strong-field optimizers of the matrix-Bingham free energy log E exp{s tr(AP)}. For 1<r<d/2, an exact cubic Haar moment and a uniform C2 perturbation theorem make the positive one-spike spectrum uniquely optimal at weak field. A parameter-uniform differentiated Grassmann Laplace expansion upgrades the strong-field Ky Fan limit to exact eventual uniqueness of the rank-r two-block spectrum. The globally optimized free energy must therefore fail to be real analytic at a finite field; at its first exit from the one-spike branch there is either nonconjugate coexistence or transverse-Hessian degeneracy. The balanced case r=d/2 is selected by a negative quartic coefficient instead. Second, we determine the unrestricted classical Shannon rate–distortion function for squared-Frobenius reconstruction of rho_P on a nonempty open interval adjacent to zero distortion. Arbitrary standard-Borel classical memories and arbitrary reports reduce by conditional least squares to the full posterior-density body, not merely to the source Grassmannian. Exact eventual two-block optimality and radial localization reduce the complete Shannon dual to one variable. The frontier is attained by a covariant matrix-Bingham channel whose posterior mean is a depolarized rank-r projector. A Jacobi–Selberg calculation gives the exact high-rate constant, and complementation transfers the result to every nontrivial rank. The complete intermediate-phase classification and the full all-distortion rank-r curve remain open.
At L boundary frequencies, a passive lossless network must route one fixed k-dimensional input channel to prescribed output k-planes. For targets with joint span dimension r, every square finite rational-inner interpolant satisfies the sharp universal sandwich r-k <= d_min <= k(L-1). Direct-sum targets therefore have exact minimum McMillan degree k(L-1); when N >= Lk this maximal-memory locus is open and dense. The paper gives an explicit positive Pick completion, a realization- and inertia-based lower bound, a family of targets that collide while retaining maximal exact degree for every nonzero opening, an exact quadratic certification margin, and a fail-closed noisy projector test. For three scalar target lines it also proves the complete phase diagram: degree zero for one common line, degree one exactly for three distinct coplanar lines, and degree two otherwise. Deterministic certificates and an independent implementation accompany the proofs.
Four balanced spin kicks along the vertices of a regular tetrahedron generatean order-dependent eight-pulse echo. We prove that all 24 echo orders have oneconjugacy class in SU(2). Haar randomization of only the global orientationtherefore gives an order-law-independent, self-adjoint random-unitary channelon every spin-j matrix algebra. Its exact eigenvalues are normalized SU(2)characters, lambda_l=U_{2l}(q)/(2l+1), with multiplicity 2l+1 and an explicitfinite-pulse architecture polynomial q=1-A/2. The maximal link-reflection-positive interval connected to zero is alpha<=pi/(4j+1), with an exact endpointequation and a finite OS Hamiltonian under strict inequality. In the doublescaling N=4j+1, y=xi/sqrt(N), the full tower converges uniformly tosinc(8xi^2u). This gives the sharp full-tower boundary xi_c=sqrt(pi/8), acritical boundary law N lambda_top -> 3pi/8-2s, an explicit scaled mass profile,and an ultraviolet heat-trace integral. The fixed-rank limit recovers the fuzzy-sphere Laplacian. Poissonization yields a genuine CPTP semigroup and an exactmany-body connected nuclear profile exp(3e^-c)-1. Symbolic and deterministicmatrix certificates accompany the manuscript.The Haar average also has explicit exact finite implementations. A classicalpositive Gauss--Legendre/trapezoidal cubature at bandlimit 4j needs(4j+1)(2j+1) order-conditioned frames, or (4j+1)^2(2j+1) frames when the sameframe list must be order oblivious.The common-class proof reduces the 24 orders to the even and odd A4 orbits.For finite implementations, a separate stability bound controls l1 weighterror, orientation displacement, and pulse perturbations; total variationagainst Haar is not used for discrete frames.
A passive lossless network may route the same k-dimensional input channel to different output subspaces at different frequencies. Every two-frequency restriction can be cheap while the joint task is not. For L distinct boundary frequencies and mutually orthogonal prescribed output k-planes, we prove that the minimum McMillan degree among square finite rational-inner transfer matrices regular at the interpolation nodes is exactly k(L-1). Each pair alone requires degree k, so the largest two-node minimum underestimates the global memory by the unbounded factor L-1. The result is independent of node spacing. Necessity follows both from a zero budget for an analytic minor and from a Pick-Stein displacement identity whose negative inertia cannot exceed realization rank. Sufficiency follows from an explicit positive Pick completion. Beyond exact orthogonality, we derive a general cross-frequency inertia certificate, a gauge-independent span bound, an open robust region preserving the integer memory count, and a fail-closed noisy-eigenvalue certificate. Deterministic code tests arbitrary nodes, synthesizes minimal conservative realizations, and independently verifies the stated identities.
Under an explicit noncontextual homogeneous-response ansatz for rank-one quantum effects, we study normalization on every unitary rotation of a fixed finite POVM. The orbit-normalization operator diagonalizes on the projective harmonics of complex projective space. We derive its weighted all-degree singular spectrum and show that the smallest non-Born spectral value is the sharp condition number controlling distance from the affine space of trace-one Hermitian quadratic responses. For dimensions at least three, the high-degree spectrum of a finite seed converges to the sum of squared aggregate ray weights; for an outcome-simple equal-trace seed with N outcomes, the limit is d squared divided by N. Combining this ceiling with the forced geometry of the two smallest tight frames solves the global equal-trace design problem in every dimension d at least five. An orthonormal-basis seed uniquely maximizes the all-degree Born-rigidity gap, with value d minus 6 divided by d plus 1, up to unitary equivalence and outcome relabeling. Every distinct-outcome overcomplete seed in the stated class is strictly worse. By contrast, every weighted complex projective 2-design has a degree-two nullspace and therefore zero rigidity gap. Thus measurements optimal for state tomography can be maximally non-rigid for this covariant normalization objective. The result concerns one-step response fields and does not derive state update, intrinsic randomness, or microscopic particle dynamics.
For a prescribed traceless Hermitian five-level target F, we determine exactly the least product of Hilbert-Schmidt norms of Hermitian A,B satisfying -i[A,B]=F. If g_i=lambda_i-lambda_{i+1} are the ordered spectral gaps, the answer is the maximum of six explicit rational linear forms and their reversals. Exact Littlewood-Richardson enumeration gives 142 Horn inequalities. Twelve sparse dual identities prove the lower facets, and twelve rational singular-spectrum maps attain them; exact cone certificates prove completeness without sampling. In contrast with the two qutrit facets and four four-level facets, dimension five has twelve exposed facets and open chambers that force rank four. For every nonzero target, the least rank of an optimal factor is max(n_+(F),n_-(F)). We also prove the sharp tax 1 <= kappa_5/(||F||_1/2) <= 5/2 with complete equality loci, and the exact balanced triangular perimeter S_2^2=12 sqrt(3) kappa_5. Two rational certificates and an independent deterministic LP campaign accompany the paper.
For a prescribed traceless Hermitian four-level target F, we determine exactly the least product of Hilbert-Schmidt norms of Hermitian A,B satisfying -i[A,B]=F. With ordered eigenvalues lambda_1 >= ... >= lambda_4, the answer is the maximum of four explicit linear spectral forms. The lower bounds are Horn-Littlewood-Richardson certificates and four closed constructions attain them. On every nonzero spectral stratum, the least optimal rank is exactly max(n_+(F),n_-(F)), closing all chamber walls and degenerations. We also prove a finite-dimensional Horn linear-program reduction, the sharp tax 1 <= kappa_4/(||F||_1/2) <= 2 with complete equality cases, and the exact balanced three-kick action S_2^2=12 sqrt(3) kappa_4. Exact symbolic and deterministic LP certificates accompany the manuscript.
A balanced sequence of short Hamiltonian kicks has no first-order drift, but its order-dependent commutator holonomy survives when the order record is discarded. We prove that a reversal-symmetric average of complete order/reverse echoes is a self-adjoint random-unitary transfer operator and obeys the nonperturbative bound cos(4|y|S) I <= T_y <= I for |y|S < pi/8. This yields site and link reflection positivity and a finite-dimensional Osterwalder-Schrader Hamiltonian. The physical transfer approaches the curvature-frame generator with an explicit sixth-order remainder, giving quantitative O(y^2) convergence of its mass gap. For the regular tetrahedral qubit loop, we derive the exact finite-pulse depolarizing eigenvalue, isolate its first positivity zero at y = 0.455698535295322..., and obtain the mass expansion. We then prove a complete finite-depth Hausdorff criterion for visible transfer edges, with uniformly sharp depth r-1, and give an exact rational four-kick certificate: the true frame gap 10/3 is certified while the false claim 4 has rational witness value -1/64. A Wishart-Loewner band supplies finite-sample type-I control for independent Gaussian correlator sketches, including adaptive witnesses. Exact symbolic and deterministic numerical certificates accompany the manuscript.
A passive lossless network pays two different costs when it rotates a signal subspace over a frequency arc: a continuous Wigner—Smith action and an integer McMillan memory. Known Grassmannian length bounds and rational-inner degree identities constrain these resources separately. For the stated finite rational-inner class, we determine their joint attainable region exactly. Let two rank-k subspaces have principal angles βj, put B = Σj βj and let r be the number of nonzero angles. For a square finite rational inner transfer matrix of degree at most d, define the arc action A = ∫I tr Q(θ)dθ, where Q = −iS*∂θS is positive semidefinite. If B > 0, the exact feasible set is d ≥ r and 2B ≤ A ≤ 2πd − 2B. Both faces are attained. Equivalently, dmin(A) = max{r, ⌈(A + 2B)/(2π)⌉}. The upper face is a return cost: the complementary arc must rotate the subspace back, while the full-circle trace action is exactly 2π times the degree. We prove sufficiency by an explicit compiler of normalized rank-one Blaschke—Potapov gates; it realizes every interior point and both faces, including rank-deficient and orthogonal cases. When the endpoint subspaces coincide, the region changes discontinuously to A ∈ [0, 2πd), with the upper endpoint open. We derive fail-closed noisy degree certificates and give a multi-frequency warning using classical boundary Nevanlinna—Pick theory: three pairwise degree-one routing tasks can require degree two jointly. For orthogonal-line data the obstruction is a frame-independent cycle phase, exhibited by an exact positive semidefinite Pick completion with spectrum (2,1,0). Deterministic code compiles random points of the diamond, constructs the degree-two colligation, and is replayed by an independent verifier.
A balanced sequence of short Hamiltonian kicks has zero first-order drift but retains an order-dependent commutator holonomy. For a traceless qutrit target with ordered eigenvalues, we solve the inverse Hermitian-commutator problem exactly: its minimum product Hilbert-Schmidt norm is the larger adjacent spectral gap, equivalently half the trace norm plus the absolute middle eigenvalue. This yields the exact minimum action of a balanced three-kick realization and identifies a middle-eigenvalue cost tax between 1 and 3/2. Among four equal-norm balanced qubit kicks, the forgotten-order curvature gap is at most `S^4/108`, with equality only for a regular tetrahedron. Its 24 orders generate exactly the six Pauli holonomies, so the finite twirl is depolarizing and an explicit inverse-cosine schedule realizes any depolarizing semigroup exactly at every finite step count. We then derive the diffusion limit from complete echoed, pinched physical words, obtaining an explicit `O(n^-1/2)` diamond-norm bound. Off-block pulses of size `y^(1+beta)` produce a sharp trichotomy: irrelevant for `beta>1`, additive at `beta=1`, and Zeno-projective for `0<beta<1`; a quantitative compression bound gives uniform convergence away from zero and an explicit initial-layer profile. The complete implementation has serial action proportional to `n^(3/4)`. Exact symbolic and deterministic numerical certificates accompany the manuscript.
A balanced ordered loop of retained Hamiltonians generates a second-order geometric Hamiltonian. We solve its higher-rank action problem and determine what remains when the order record is erased. For every matrix dimension and every number of kicks `m >= 3`, a gauge-invariant operator action obeys a sharp, dimension-free polygonal bound on the spectral diameter of the geometric Hamiltonian; regular two-level polygons attain the constant in every dimension. A target-sensitive Hilbert--Schmidt companion bound is controlled by the trace norm and has equality exactly for sign-paired nonzero spectra. These laws yield exact diamond-norm ceilings with a physical state witness. For a uniformly forgotten order, we derive an exact `1/12` commutator-frame covariance. The resulting reversal-symmetric random-unitary product converges in diamond norm at rate `O(n^-1)` to a GKLS frame Laplacian whose fixed algebra is the joint commutant of the pair commutators. A tetrahedral Pauli architecture gives exact primitive depolarization with gap `64/3`. Finally, fixed laboratory time `T` and bounded kick amplitude `Lambda` impose a sharp curvature ceiling proportional to `T^2 Lambda^2/n`; maintaining nonzero holonomy requires `Lambda=Omega(sqrt(n))`. Exact symbolic and deterministic numerical certificates accompany the paper.
Lower bounds do not determine which resource profiles are physically attainable. We close the continuous inverse problem for passive subspace transport. Let two rank-k subspaces in C^N, N >= 2k, have ordered canonical angles beta, and let q_j be the integrated leading eigenvalues of a positive generator Q = -iS* dS/dt. We prove the exact equivalence: q is attainable if and only if 2 beta is weakly majorized by q. Every feasible profile has a compiler using at most k+1 mutually commuting positive generators of rank at most k, with constant ordered top spectrum. Analytic rationality adds an integer-valued resource: for two boundary frequencies, the least McMillan degree of an inner matrix sending a fixed input subspace to two prescribed output subspaces equals the number of nonzero canonical angles. We also give a fail-closed finite-error certificate for both resources from noisy projectors.At multiple frequencies, Wigner-Smith delay is only the block diagonal of a boundary de Branges-Rovnyak Pick matrix. This positive matrix is the Gram matrix of frequency-excited internal states and has rank bounded by the McMillan degree. Its spectrum yields robust degree certificates and unavoidable model-reduction tails. An exact example exhibits identical local proper delays for degree-one and degree-two devices while their Pick ranks distinguish them. Deterministic code reconstructs the compilers, tests 2,240 randomized theorem interfaces, checks the strict separation, and independently replays the certificate.
The order of short operations is normally a microscopic detail. We show that a fixed degenerate measurement turns it into an operational curvature with an exact macroscopic readout. Let E retain a block matrix algebra, and let two words apply the same short Hamiltonian kicks in different orders, with the same nonselective pinching after every kick. Their leading difference is `-i tau^2 ad(F) E`, where F is the sum of `i[h_j,h_k]` over pairwise inversions. This gives a discrete non-Abelian Stokes law. Its diamond-norm coefficient is exactly the largest spectral diameter of a block of F, and therefore fixes the leading optimal channel-discrimination advantage. Under first-order balance, all permutations have the same dissipator while their geometric Hamiltonians differ exactly by F, so local curvature integrates into distinct diffusive quantum Markov semigroups. We characterize contextual invisibility, the rank-one classical boundary, and inverse-success amplification under postselection. Every retained Hamiltonian modulo the block center is realizable by a balanced three-kick loop. For a qubit block we prove the sharp action law `kappa <= S^2/[m tan(pi/m)]`, attained by regular planar kick polygons. An integer-Pauli four-kick architecture saturates the bound and supports primitive dissipation. Symbolic, semidefinite, and convergence certificates accompany the paper.
Repeated projective measurements are usually discussed either in the ballistic Zeno scaling or after reduction to classical outcome probabilities. We study a different regime in which each nonselective measurement retains a full matrix algebra inside every degenerate outcome block. Let E(X)=sum_a P_a X P_a, let E_tau be its rotation by exp(-i tau H), and close one cycle by Phi_tau=E E_tau E. We prove, uniformly on compact time intervals and in diamond norm, that Phi_{sqrt(t/n)}^n converges to exp(-tK)E, where K=E ad_H(1-E)ad_H E=C_H^* C_H. The limit is a genuinely quantum Markov semigroup on the direct sum of the block matrix algebras, with explicit jumps sqrt(2) P_b H P_a. Beyond upper bounds, we identify three exact local obstructions for the unprocessed physical product. The cubic map M_H produces a nonzero n^{-1/2} coefficient; after it vanishes, the quartic product obstruction J_H produces a nonzero n^{-1} coefficient; after both vanish, the quintic obstruction P_H produces a nonzero n^{-3/2} coefficient. If all three vanish, the error is O(n^{-2}). All three coefficient transforms are injective. An exact three-record algebraic example has M_H=J_H=0 but P_H nonzero, proving that the third branch is attained. The fixed algebra is A intersect {H_off}', and the least singular value of the commutator frame C_H defines a noncommutative connectivity gap. In the primitive case that gap is the exact exponential mixing exponent in diamond norm and is Lipschitz robust under Hamiltonian perturbations. Rank-one blocks reduce exactly to twice the squared-coupling graph Laplacian. An eight-dimensional rational architecture with four qubit blocks is certified irreducible; its gap is the unique root in (0.4545,0.4547) of an explicit quartic. Symbolic certificates, semidefinite diamond-norm replays, and convergence tests accompany the paper.
A subspace quantum speed limit usually records only the largest canonical angle. Passive scattering theory usually records either the largest proper delay or their total trace. Both compressions discard how a rank-k routing event is distributed across modes. We derive the missing vector law: the integrated spectral-spread vector weakly majorizes twice the canonical-angle vector. More strongly, for every symmetric gauge Φ and every pair of k-subspaces in N ≥ 2k dimensions, we solve the variational problem exactly: the minimum spectral-spread action is 2Φ(β). The same value holds for the leading-proper-delay action under positive generators, and one constant coupled-mode path minimizes all gauges simultaneously. The Ky Fan members recover the bandwidth limit, strengthen the trace-action law, and expose every intermediate modal budget. An exact nonnegative slack decomposition separates common-mode delay, inefficient spectral coupling, and nongeodesic subspace motion. Robust corollaries convert heterogeneous pass/stop leakage spectra and finite tomography errors into certified Lorenz curves for delay and, for rational inner networks, McMillan-degree lower bounds. A deterministic artifact tests 8,192 random generators, 1,536 time-dependent paths, sharp equality families, and 2,048 noisy tomography instances.
Exact interpolation can force the internal degree of a passive network, but laboratory calibrations are approximate. We prove a sharp frequency-domain speed limit requiring neither exact zeros nor analytic continuation away from the measured frequency axis. Let (S(e^{itheta})) be an absolutely continuous unitary scattering path with positive Wigner—Smith generator (Q(theta)=-iS(e^{itheta})^astpartial_theta S(e^{itheta})succeq0). If a fixed (k)-dimensional input subspace is routed approximately between complementary output sectors with amplitude leakages (varepsilon_p,varepsilon_s), define (alpha=[pi/2-arcsinvarepsilon_p-arcsinvarepsilon_s]_+). Every transition then requires Wigner—Smith trace action at least (2kalpha) and largest-proper-delay action at least (2alpha). Costs add over disjoint frequency arcs. For a rational inner network of McMillan degree (n), (M) alternating pass/stop pairs imply (nge 2Mkalpha/pi), recovering (nge Mk) at zero error. An explicit (2k)-port interferometric family attains the bounds for every admissible error pair. A tomography-error corollary converts finite scattering measurements directly into certified degree and delay lower bounds. Reproducible certificates audit equality cases, positive-block inequalities and random Blaschke—Potapov products.
We study whether passive reservoir filtering can suppress decoherence more effectively when its channel mixing is irreducible, even after fixing the passband responses and rational complexity. For every integer S ≥ 5, we construct an explicit causal inner six-port network with three signal and three vacuum-loss ports. Its signal block is a rational Schur transfer satisfying 3(S−1) delayed full-spark tangential calibrations and exhibiting exponentially small leakage on two stop arcs. In contrast, every transfer of the same bidegree possessing a constant nontrivial reducing channel and satisfying the same calibrations retains unit stopband norm.We strengthen this exact separation with a quantitative finite-error obstruction: for calibration defect δ and sampled reducing-line defect β, comparator leakage is bounded below by [1−C_S(δ+β)]_+, with C_S given explicitly by finite singular-value margins. A scalar Schur construction proves that every bound of this form must deteriorate at least as 2 exp(9S/20)(1+o(1)); hence uniform robustness is impossible for the chosen clustered calibrations.For uniformly nondegenerate bath spectra, the signal-level separation is squared at the Kossakowski-rate level. A closed Markov pure-dephasing model includes all auxiliary vacuum ports exactly, producing an architecture-independent measurable baseline and an explicit total Ramsey-rate advantage. Finally, we prove that strong passive suppression requires large dwell time: under a peak-delay budget D, the rate-improvement factor is asymptotically at most quadratic in D/S. The construction therefore moves the coherence-maintenance resource into passive memory, vacuum noise and conditioning rather than eliminating it. All certificates, figures and numerical audits are publicly reproducible.
Scalar Chebyshev filtering treats every vector in a block Krylov iterate with the same polynomial, while simultaneously diagonalizable matrix coefficients amount to independent scalar filters after a fixed channel rotation. We study the larger class of Hermitian matrix-polynomial filters under tangential pass constraints. Already at fixed channel dimension d=3, we construct rationally generated signatures with quantitative full-spark margin for which an irreducible symmetric filter has stopband leakage O(exp(-cN)). In contrast, every exactly calibrated symmetric coefficient family with any common nontrivial invariant channel subspace has leakage at least one; this comparator strictly contains the pairwise-commuting class and permits a fully noncommutative 2 by 2 block. A quantitative theorem covers approximately reducible filters by charging their sampled off-block coupling together with calibration error. The signature margin and the admissible combined error are both exp(-O(N)), not exp(-N^3). This exponential scale is unavoidable: for arbitrary pass nodes and signatures in the same separated bands, an explicit scalar binomial-tail polynomial has both pass error and stopband leakage at most exp(-2N/81). Thus constant-error separation is impossible, while the irreducible construction achieves the correct exponential scale class. An affine cosine substitution gives the same robust law for fixed-latency reciprocal linear-phase MIMO FIR filters. We also give an exact rational five-tap certificate with stopband norm at most 25/32 and prove that its advantage survives reducible calibration errors delta < 7/1920. A second exact-arithmetic certificate is calibrated from a public triaxial pump-vibration data set: it gives leakage below 0.96 versus one for every exactly calibrated reducible symmetric class, while its maximum directional error on six held-out records is 0.00385. In a frequency-domain-decomposition test on the frozen held-out cospectra, filtering rotates the leading modal direction by at most 0.222 degrees and changes its leading spectral ordinate by at most 2.4 × 10^-5 relatively. Numerical programs test the mechanism and expose an ordinary graph-denoising setting in which scalar Chebyshev filtering is instead preferable. Tangential matrix interpolation, MIMO filtering, scalar two-band approximation and convex FIR design are not claimed as new; the contribution is the fixed-order reducible/irreducible separation together with its two-sided calibration scale.
For the spatial Z_2 (Ising-slice) system inside the Dobrushin window 2 tanh|beta| + 2 tanh|gamma| <= alpha < 1, we machine-check in Lean 4 an end-to-end chain from the Gibbs measure to the spectrum of the reconstructed transfer operator. (i) The Osterwalder-Schrader (site-form) reconstruction of the transfer operator is unitarily conjugate, by the explicit sqrt(w) boundary dressing, to the symmetrised Dobrushin kernel. (ii) The unnormalised Gibbs sums themselves are exact matrix elements of that operator's powers: gibbsPathSum(w,beta,N,A,B) = lambda^N , with the partition function the same shape at the dressed constant. These are identities, not bounds, and they hold at every real beta and every positive weight. (iii) There is one mass m > 0 such that for every spatial extent L the projected operator norm is at most e^{-m} and every mixed connected correlator obeys | - | <= ||u|| ||v|| (e^{-m})^n, the zero-time case included. (iv) The connecte two-point function of the normalised Gibbs measure decays at that same rate with a constant independent of the time depth; dividing by the partition function is licensed by a denominator floor uniform in N, which the positive cone supplies and the spectrum does not, since the spectral route controls only the even powers. (v) The N -> infinity limit state exists, is the vacuum state of the reconstructed operator, and does not depend on the strictly positive observable terminating the chain. (vi) The reconstructed operator is a reversible Markov chain -- stochastic and in detailed balance for pi = Omega^2, both proved -- and in that stationary state the connected correlator of bounded observables obeys |E_pi[f P^N g] - E_pi[f] E_pi[g]| <= K_f K_g (e^{-m})^N, with quantifier order "there exists m, for all L": no factor depending on the spatial extent. Summing over time separations gives a susceptibility bound K_f K_g / (1 - e^{-m}), independent of the cut-off and of the extent. The window is non-empty at an interacting point (beta = gamma = 1/10, alpha = 1/2), machine-checked, so none of these conditionals is vacuous. The analytic input is inherited: the mass is the one the Dobrushin corollary already produced, and the window is not widened. What the reconstruction contributes is the identification, the exact identities, and the normalisation in which both the rate and the constant lose their dependence on the volume.
We determine when rapid exact maintenance of a rank-deficient quantum target produces logarithmically divergent free-energy restoration power in algebraic quantum field theory. On every sigma-finite properly infinite factor we construct a bounded quantum Markov semigroup, a nonfaithful normal target and a faithful invariant reference state for which Araki relative entropy reduces exactly to a binary divergence. We derive the semigroup first from localized thermal fermionic probe collisions and then from one autonomous finite-bandwidth Dirac KMS reservoir with fixed smooth coupling. Exact memory equations yield an explicit finite-coupling Davies bound on finite van Hove windows. For a displayed compactly supported massless Dirac form factor, Araki—Wyss regularity, threshold behaviour and Fermi-golden-rule positivity give a completely bounded Davies approximation uniformly for all times with error of order (O(|lambda|)); the sharper (O(lambda^2)) result is isolated under additional reduced-resonance hypotheses. We construct a background-covariant two-Dirac-field completion using Green operators, Møller maps and relative Cauchy scattering, proving naturality, causal factorization and exact spacelike triviality. A locality obstruction shows why a strictly local multiplier cannot coincide exactly with the solvable rank-one reservoir coupling, while a Feshbach reduction quantifies the correction. Finally, we prove that no fixed faithful vacuum or KMS restriction can exhibit the rank-boundary mechanism, but faithful families with a vanishing spectral floor recover its complete coefficient. The results separate algebra type, target faithfulness, microscopic realizability and regulator uniformity, and provide reproducible numerical audits of the finite-dimensional identities and explicit Dirac form factor.
We present a Lean 4 formalization of a thermodynamic-limit construction for the anisotropic nearest-neighbour Ising model on the two-dimensional integer lattice. A telescoping comparison argument and the classical Dobrushin resolvent produce a volume-uniform expectation bound. Exact restriction and reindexing maps between finite Gibbs measures yield convergence of the complete free-boundary sequence, independence of auxiliary envelopes and cofinal samplings, stability under receding boundary perturbations, and equality of free and periodic limits.Finite-support cylinder presentations are quotiented by equality of their represented functions on the full spin space. The resulting local algebra carries a genuine lattice-translation action, and the limiting functional is positive, normalized, real-linear, and invariant under every integer translation. We equip this algebra with its intrinsic uniform norm, construct its complex star-algebra representation, take the corresponding commutative C*-closure, and extend the limiting functional to a positive complex-linear functional of norm one. We also formalize normalized Gibbs conditional kernels for arbitrary finite conditioning sets and prove positivity, exterior locality, idempotence, and the exact finite-volume Gibbs tower identity.All results remain within the classical anisotropic Dobrushin region. The infinite-volume DLR fixed-point equation for the completed state is not claimed.
We study whether a flat lattice-gauge Hodge form supplemented by a penalty depending only on a line-integral block variable can control the full fine one-cochain norm uniformly in the block side. The constant-field sector selects the exponent (d-2)/2 only under exact scale neutrality, but this scaling test does not address the kernel of the block map. We construct transverse, divergence-free, block-periodic Fourier cochains that are annihilated exactly by the block map. Their first-mode Hodge Rayleigh quotient is 4 sin²(pi/L), forcing every admissible full-space Poincaré constant to grow at least quadratically with L. The obstruction applies to every scale-dependent scalar functional factored through the block variable and satisfying only that it vanishes at the coarse zero field; no linearity, continuity, positivity, or growth condition is required. For linear postconditioners, higher Fourier frequencies yield an orthogonal real subspace of dimension 2R and a min—max bound on the 2R-th eigenvalue, establishing a growing low-energy spectral cluster. We also derive an exact repair identity and a necessary witness-channel budget for modified block measurements and additional fine-space terms. The kernel construction, Hodge energy, factorized no-go theorem, repair identity, and necessary repair budget are formalized in Lean 4. The result concerns a finite periodic flat full-domain form and makes no claim about interacting coercivity, gauge quotients, continuum limits, or the Yang—Mills mass gap.
