Research paperAcceptedARR-2026-7NPRNBW4488HG90K · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-1D2QV1RP1292JREW · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-3H0ZKWJMH18MH9FX · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-7D2BBEC8MJ8BM80S · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-15SJ1ANHDN8D88Z1 · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-263B0753CQ9J2T34 · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-6F8XRSBM0J9Q2R2B · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-7X5XX0CBE19MXVSZ · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-7CCV86W3Y59VS8PN · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-6DJ302B1G38V4SHD · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-6WX2JF38WE87GB2M · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-6XH6JAS5ZA934A6J · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-5KS70GV7KK9DYA69 · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-7H9FAPTBZA897AMJ · v2
Lluis Eriksson
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 a scalar radial free energy with q=n-2 and overlap normalizer L_q(kappa)=q! sum_{j>=0} kappa^(2j)/(q+2j)!. We solve the remaining global phase problem in every dimension. For the Turanian-like combination J_q=L_q L_q' - kappa L_q L_q'' + kappa(L_q')^2, every coefficient is expressed through the variance of a central parity-truncated binomial law. Likelihood-ratio ordering and an exact uniform tail bound 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 an explicit linear face followed by one covariant branch for every n>=6; together with the continuous cases n=3,4,5, this completes the all-dimensional phase diagram. We also derive sharp large-dimension expansions for the coexistence field, active radius, distortion, and information, including the logarithmic finite-size displacement. As operational corollaries, the same RDF equals the minimum worst-source channel capacity, arbitrary joint N-letter classical memories cost N R_n(D), and the iid Haar source has constant tilted information and zero dispersion. The N-letter identity is a mutual-information/capacity theorem, not an exact finite-codebook achievability result. The work concerns signed Pluecker-coordinate compression, not quantum rate–distortion or Born-probability prediction.
Research paperAcceptedARR-2026-6M3VGTXZ6W8JW9C9 · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-7XT7AB8WJP9QPTGT · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-61Y0FFA39M8KMBJ5 · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-77QM18J2KG9679B7 · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-52B6MSS1W197W9T2 · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-1D2QYXPCVY9H7ANB · v1
Lluis Eriksson
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.
Research paperAcceptedARR-2026-6FDEKPVJ0W8BHBMC · v1
Lluis Eriksson
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.
Research paperHistorical importARR-2026-33BE6K3HW78F4T7F · v1
Lluis Eriksson
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.
Research paperHistorical importARR-2026-1WXCVX96S68GA9VK · v1
Lluis Eriksson
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.
Research paperHistorical importARR-2026-0DZQ2WNKPW8GCTPA · v1
Lluis Eriksson
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.
Research paperHistorical importARR-2026-37B8R0QTA894GTFF · v1
Lluis Eriksson
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.
Research paperHistorical importARR-2026-1CYF5ZA33T9A2AD6 · v1
Lluis Eriksson
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.
Research paperHistorical importARR-2026-1PEAQQ7JWA8X19W4 · v1
Lluis Eriksson
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.
Research paperHistorical importARR-2026-6EVZ1F28PK8YDR6X · v1
Lluis Eriksson
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.
Research paperHistorical importARR-2026-51CS1EZ4K28W7TB3 · v1
Lluis Eriksson
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.
Research paperHistorical importARR-2026-7EPM3V3DP796BBRR · v1
Lluis Eriksson
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.
Research paperHistorical importARR-2026-3FHSR0Q6W49X38F6 · v1
Lluis Eriksson
For beta > 0 let I_m = I_m(beta) denote the modified Bessel functionof the first kind and define a_m = I_m^2 ((m-1) I_(m-1)^2 + (m+1) I_(m+1)^2), b_m = m I_m^4,with sine series F_A(t) = sum_(m>=1) a_m sin(mt) andF_B(t) = sum_(m>=1) b_m sin(mt). The associated Wronskian is negativeexactly when F_A/F_B is decreasing. We isolate what can and cannot beproved from two natural inputs: the Neumann convolution kernelI_0(2 beta sin(phi/2)) and the small-coupling anchor whose normalizedlimit is 4 sin^3(t).First, the convolution does prove F_B(t) > 0 for 0 < t < pi. Second,the endpoint is governed by two alternating quantities, c_3 and B_pi,through an exact cubic law. We prove B_pi > 0, derive integralrepresentations, and establish that the cancellation lost by replacingthe alternating quantities with positive-term majorants has exponentialrate 8 - 4 sqrt(2). Finally, we prove a smooth one-parameter perturbationtheorem: one may keep F_B and its kernel unchanged, preserve positivityand strict coefficient-ratio ordering, and preserve every jet at beta = 0,while choosing either sign of the endpoint cubic coefficient.Consequently those structural data, even taken together, do not implyglobal Wronskian negativity. This is a no-go theorem for a proofarchitecture, not a counterexample to the original Bessel conjecture.A short high-precision kill-test accompanies the paper; nocomputer-assisted inequality is used in the proofs.
Research paperHistorical importARR-2026-3BQ8ENXN868SKV43 · v1
Lluis Eriksson
A formal Lean 4 development revisits a two-dimensional spatial transfer kernel whose earlier uniform spectral argument relied on constant row sums. We show that the loss of row-sum constancy obstructs that method, not the conclusion. With no sorry and no project axiom, we prove twelve machine-checked theorems.The chain establishes the sharp field-uniform one-bond influence envelope tanh J; assembles these bounds into a finite-volume Dobrushin matrix; proves a volume-independent resolvent estimate under row-sum bounds alpha < 1 without assuming constant rows; mechanises Dobrushin's comparison inequality with the attained 1/4 covariance constant; constructs Gibbs measures, heat-bath kernels and intrinsic influence matrices from arbitrary strictly positive finite weights; and specialises to anisotropic Ising interactions. For L x T rectangles with free boundary, the condition 2 tanh|beta| + 2 tanh|gamma| <= alpha < 1 yields exponential decay of correlations with beta, gamma, alpha and the prefactor fixed before the volume quantifiers.The transport into the operator formulation is closed: an exact finite band identity relating endpoint covariances to matrix elements; an abstract theorem showing that a common exponential decay rate for band covariances - a hypothesis on finite path measures, carrying no operator, norm or spectrum - implies a uniform positive gap for a family of projected transfer operators; a Perron-boundary tilt identity that preserves the decay rate while absorbing boundary costs into extent-dependent constants; an exact currying identification of the free strip measure with the rectangle Ising measure; and the resulting corollary: inside the window there is one m > 0 bounding the projected transfer operator of the coupled kernel's normalised Perron data by exp(-m) at every extent, with m = -log alpha.Numerical measurements at L <= 12 provide counterevidence to attributing spectral degeneracy solely to nonzero spatial coupling. No infinite-volume state, thermodynamic limit or boundary-condition independence is constructed; the window is sufficient, not sharp. The underlying Dobrushin mathematics is classical; the contribution is a non-vacuous, reproducible mechanisation and composition of the full chain, ending at a volume-uniform operator gap. No consequence for Yang-Mills theory is claimed.
Research paperHistorical importARR-2026-627PF1B4KK9VTRYT · v1
Lluis Eriksson
Let a fine periodic lattice have side LN' and let Q_L be the L^{-d}-normalized block average of length-L line integrals. In four dimensions, the rescaling Q_L -> LQ_L repairs the elementary constant-field scaling obstruction to a Poincare estimate. We prove that it cannot yield coercivity on the full one-cochain space with a constant uniform in the block side.For every L >= 2, every fixed N' >= 1, and N_c >= 2, we embed the first within-block Fourier phase zeta_L = exp(2 pi i/L) in a real two-plane of the internal coordinate space and construct a transverse one-cochain A_L. It satisfies Q_L A_L = 0 and div A_L = 0. For every dimension d >= 2,||A_L||^2 = (LN')^d, = (LN')^d lambda_L,where lambda_L = 4 sin^2(pi/L) <= 4 pi^2/L^2.Thus the Rayleigh quotient of K_0 + (s_L Q_L)^*(s_L Q_L) is exactly lambda_L for every scalar rescaling s_L. Every admissible full-space Poincare constant obeys C_P(L) >= 1/lambda_L >= L^2/(4 pi^2), so no constant chosen before L can work. Here "volume-uniform" means uniform while the block/fine side L varies at an arbitrary fixed positive coarse side N'; it is not a claim about N' -> infinity at fixed L.The construction, kernel identities, exact Hodge energy, Rayleigh identity, and quantified no-go theorem are formalized in Lean 4. The focused axiom audit reports only propext, Classical.choice, and Quot.sound. The theorem concerns the stated flat full-domain form; it does not contradict gauge-restricted propagator constructions and makes no infinite-volume, continuum, or Yang-Mills mass-gap claim.
Research paperHistorical importARR-2026-5V5K6T6DQT9J8SY6 · v1
Lluis Eriksson
We machine-check the exact measure and differential bridge between thefour-edge SU(2) crossing chart and the two effective group coordinates used bya finite-dimensional Ward identity. The quotientr(a)=(a2 a4^-1,a1 a3^-1) pushes normalized four-fold Haar measure exactly tonormalized two-fold Haar measure. Two gauge-compensated flows on the originalfour edges intertwine with independent right multiplication of the quotientcoordinates, so first and mixed derivatives of the crossing Wilson worddescend without choosing a gauge. The three Pauli directions are verifiedindividually, their mixed generators close into direct and reverseresolutions, and the two-coordinate Ward theorem lifts to a literal four-edgeintegral with coefficients -1/4 and -1/2. This compensated operator is notidentified with the ordinary-edge mixed operator of Driver-Hall-Kemp: thepaper writes both operators and their distinct reverse resolutions and markstheir comparison as open. The Lean producer has 26 public declarations and 16audited theorems. No weak four-face heat-kernel identity, area derivative, orfull Makeenko--Migdal equation is claimed.
Research paperHistorical importARR-2026-0JTZHGRV1C8G98TM · v1
Lluis Eriksson
We formalize extended gauge invariance at a simple four-edge SU(2) crossing and connect the geometric edge chart to the two-coordinate chart used by a machine-checked crossing Ward identity. On SU(2)^4 we define the two opposite-edge right actions from the abstract Makeenko--Migdal theorem, prove that they are commuting product-Haar-preserving actions, and show that their common parameter composes to ordinary vertex gauge invariance. The four-edge Wilson word tr2(a3^-1 beta a2 a4^-1 alpha a1) is proved invariant under both half-actions. We construct the explicit quotient r(a)=(a2 a4^-1,a1 a3^-1), a canonical section, and prove existence and uniqueness of the universal factorization for every extended-gauge-invariant complex function. The complete map from the cyclic four-edge chart to physical and gauge coordinates is proved to preserve literal four-fold Haar measure in one public endpoint. Finally, the four-edge Wilson word is identified exactly with the prior two-coordinate crossing word evaluated on r(a). The Lean producer has 56 public declarations in 488 physical lines; all 36 theorems depend only on propext, Classical.choice, and Quot.sound, with no local proof escape. No heat-kernel area derivative or full Makeenko--Migdal equation is claimed.
Research paperHistorical importARR-2026-3B7MWNSJS18TYAA4 · v1
Lluis Eriksson
We formalize a finite-dimensional crossing Ward identity for fundamental SU(2) Wilson words. On every finite edge space SU(2)^E, two distinct coordinates are inserted into a concrete normalized trace word. Lean proves entrywise that right multiplication by each of three explicit SU(2) curves produces the normalized Pauli generators i sigma_a/2, differentiates the trace word once at each selected coordinate, and identifies the Pauli-summed mixed derivative with the rank-two Fierz contraction. Two applications of Haar integration by parts transfer the corresponding mixed generators from a density to the Wilson word. The resulting integral closes exactly on the direct and reverse single-trace resolutions, with coefficients -1/4 and -1/2 in the chosen normalization. The producer contains 19 public declarations in 487 physical lines; all 13 new theorems depend only on propext, Classical.choice, and Quot.sound, with no local sorry, admit, or axiom. This is not a full Makeenko--Migdal area equation: the remaining physical input is a weak four-face identity against crossing-certified extended-gauge-invariant observables. In the program's heat time, generated by the Pauli Laplacian, its coefficient is kappa = 2.
Research paperHistorical importARR-2026-7S1XA3PZEB9QM8FQ · v1
Lluis Eriksson
A positive site weight acts on a transfer kernel by congruence, K -> DKD with Dpositive diagonal, not by similarity. Similarity preserves the entire spectrum;congruence preserves strictly less, and strictly more than nothing. We determineboth halves for the kernel of L decoupled Ising bonds. Rigid half: the sign ofevery quadratic form value survives, so definiteness in either sign is acongruence invariant -- the definite case of Sylvester's law of inertia, in aform a machine can check without diagonalisation. Fragile half: the subdominantratio r (second eigenvalue modulus over the Perron root) does not survive atall, and we locate exactly how far it moves. Restricting the L-site kernel tothe two antipodal configurations leaves one Ising bond of coupling bL, of ratiotanh(bL) for b>0: a weight concentrating there fuses the L sites into a singleeffective site carrying L times the coupling. Since tanh(bL) -> 1, no boundr <= rho < 1 holds simultaneously in L and over the whole positive-diagonalcongruence orbit; the obstruction to volume-uniformity is a property of thecongruence, not of any spectral estimate, so an argument bounding r throughcongruence invariants alone cannot produce an L-uniform bound. For arbitraryweights we prove sup_{D>0} r(DMD) = (1-m)/(1+m), where m is the leastoff-diagonal entry: the least correlated pair determines the supremum. (Weevaluate the supremum; we do not classify its maximisers.) The proof isgeometric rather than spectral -- Hilbert's projective diameter is itself acongruence invariant, and Birkhoff's contraction theorem converts it into thebound -- and uses no definiteness, only positivity of the entries. The lowerbound uses no limiting argument about spectra: a two-supported fluctuationvector gives the estimate at each strictly positive epsilon, so no continuity ofeigenvalues is imported anywhere.
Research paperHistorical importARR-2026-2FKPJ10YES8A0TWE · v1
Lluis Eriksson
We formalize the finite-dimensional integration-by-parts mechanism used inlocal proofs of the Makeenko—Migdal equation. For a measure-preserving realflow on a finite measure space, Lean verifies differentiation under theintegral from an explicit locally uniform integrable majorant and proves thatthe integral of the generator vanishes. A bounded dominated product interfacethen yields integration by parts, and two successive applications transfer amixed pair of generators from a density to an observable with positive sign.These results are instantiated on normalized Haar probability measure ofconcrete SU(2) and on every finite product SU(2)^E, for left and rightmultiplication of one selected edge. To fix the representation normalization,we construct three explicit trigonometric curves in SU(2) and prove entrywisethat their tangents at the identity are i sigma_1/2, i sigma_2/2, and isigma_3/2. The producer contains 30 public definitions, structures, andtheorems in 519 physical lines; its audit contains no local sorry, admit, oraxiom, and the headline results depend only on propext, Classical.choice, andQuot.sound. This closes the Haar integration-by-parts layer. It does not claimthe full four-area Makeenko—Migdal identity: the remaining formal inputs arethe heat-density directional identity, extended gauge invariance at acrossing, and their geometric assembly.
Research paperHistorical importARR-2026-4H8JZ663XA9XCBMN · v1
Lluis Eriksson
We give a kernel-checked positive-area calculus for the concrete SU(2) classheat kernel used in two-dimensional Yang—Mills theory. With irreducible labeln, dimension n+1, and Casimir c_n=n(n+2)/4, Lean verifies at every positivetime and every derivative order that the infinite spectral jet converges anddifferentiates term by term to the next jet. The hierarchy is packaged as aC-infinity map on the positive-time half-line, using explicit uniform summablemajorants on positive-time neighbourhoods. We then differentiate the literalnormalized-Haar two-face Migdal integral, prove that its left and right areaderivatives equal the first spectral jet at the merged area, and showinfinitesimal invariance under (s,t) -> (s+u,t-u). Finally, every normalizedWilson character satisfies its exact Casimir area ODE as an actual Haarintegral against the infinite heat kernel. The artifact contains 25 publicdefinitions and theorems, no local placeholders, and audited dependencies onlyon propext, Classical.choice, and Quot.sound. We do not claim the four-faceMakeenko—Migdal crossing equation; the remaining inputs are local Lie-groupintegration by parts and certified crossing geometry.
Research paperHistorical importARR-2026-7C7E7PRA1Y8MNAWK · v1
Lluis Eriksson
Finite-rank Makeenko-Migdal equations generate products of Wilson traces at self-intersections. For SU(2), this apparent multitrace obstruction closes exactly on single traces, but the statement is normalization-sensitive: the traceless Lie algebra contributes a finite-rank correction that disappears for U(2) and must not be dropped. We give a Lean 4/Mathlib formalization of the complete group-algebraic closure mechanism on Mathlib's concrete special unitary matrix group. With normalized trace tau(A)=Tr(A)/2 and normalized anti-Hermitian Pauli directions X_j=i sigma_j/2, the kernel checks the Casimir identity, the rank-two Fierz identity, the induced crossing contraction, and the SU(2) trace-skein identity tau(g)tau(h)=(tau(gh)+tau(gh^{-1}))/2. Consequently, the finite-SU(2) crossing term tau(g)tau(h)-tau(gh)/4 equals tau(gh)/4+tau(gh^{-1})/2. We then formalize a universal local interface with four cyclically ordered branch holonomies, an independent orientation on each branch, the two opposite-strand words, and precisely the two direct/reversed reconnections. Its corrected crossing term closes on those reconnections for every branch assignment and orientation choice. A recursive theorem also extends the reduction to products of arbitrarily many fundamental traces. The identities are classical; the contribution is a concrete, kernel-checked normalization bridge from Pauli contraction to the single-trace closure used in finite-rank loop equations. We do not claim a formal derivation of the Yang-Mills area derivative, planar loop geometry, or the full Makeenko-Migdal equation.
Research paperHistorical importARR-2026-223XBWQ8PQ8Z2RDS · v1
Lluis Eriksson
BOND REFLECTION, AND THIS ONE IS ABOUT THE MEASURE. A whole path X of 2m+2slices has a past half past(X) and a future half; write rev(X) for the futurehalf READ BACKWARDS FROM THE FAR END, which is the reflected copy. Let F be acomplex observable of an entire half. Then for every L, every m, everystrictly positive source weight w and every beta >= 0, sum over whole paths X of conj(F(past X)) * F(rev X) * W(X) >= 0the sum running over all whole paths and W being the ordinary Gibbs weight.This is the finite-volume Osterwalder-Schrader reflection-positivityinequality for this model at odd separation. The sum is UNNORMALISED, which isthe form the axiom is about; dividing by the partition function is division bya positive number and preserves the sign. The Gram matrix of a finite familyof such observables, with complex coefficients, satisfies it as well. It isobtained by proving that assembling a past half, a crossing bond and areversed future half is a BIJECTION onto paths, and that the weights multiplywith exactly one crossing factor.SITE REFLECTION, AND THIS ONE IS NOT, YET. Through a site the two halves SHAREthe middle slice, so the assembly is not a product of two independent halvesand its bijection is a different statement. What is proved there is thehalf-chain form, its collapse identity and its Gram positivity -- at EVERYbeta, negative coupling included -- but NOT its identification with the pathmeasure. For that geometry the object remains a candidate, and the statustable says so row by row.ON THE BETA HYPOTHESIS. beta >= 0 is proved SUFFICIENT and is not provednecessary here; at L = 0 it could not be, since the bare kernel is then thescalar 1. The witness that would make the boundary exact from one site upwardsis authorised by a gate and is not written.THE MECHANISM, IN ONE SENTENCE. Summing out the interior of a half sends F toa vector indexed by its boundary slice alone, after which the bond casereduces to positive semidefiniteness of the kernel -- machine-checked in thecompanion paper -- and the site case to a weighted squared norm, which uses noproperty of the kernel at all.WHERE THE TWO REFLECTIONS DIFFER, AND IT IS NOT COSMETIC. Through a bond thetwo halves are disjoint and meet through one kernel factor, so positivityneeds the kernel itself positive semidefinite -- which, for L >= 1, holdsexactly when beta >= 0. Through a site the halves SHARE the middle slice; theform is then a sum of squared moduli divided by the weight of that slice, soit is non-negative for EVERY beta, negative coupling included, and no propertyof the kernel is used at all.WHAT THE SOURCE WEIGHT DOES. In the companion paper the weight was handled bycongruence: conjugation by sqrt(w) cannot change the sign of a quadratic form.Here it is not conjugated away, it is SUMMED away -- and what is left overdiffers between the two geometries. In the bond case every source weight isabsorbed into the collapse and the bare kernel remains; in the site case theshared boundary slice survives as a factor 1/w(sigma), because each halfcarries that slice's weight and the product would count it twice. Differentreasons, same conclusion: nothing in the hypotheses depends on w beyondpositivity.WHAT THIS IS NOT. No reconstruction: the physical Hilbert space as thequotient of the past algebra by the null space of this form is not built.Nothing here concerns uniformity in the extent, SU(N), the continuum limit, orthe Yang-Mills mass gap.ON THE PRE-REGISTRATION. Four gates were committed before a line of the modulewas written; the status table reports what happened to each. Two of them reada minimum eigenvalue rather than sampling observables, which is the instrumentthe previous campaign's autopsy said was needed.
Research paperHistorical importARR-2026-02YRWX53DG9M3R0R · v1
Lluis Eriksson
WHAT IS PROVED, STATED BEFORE ANYTHING ELSE. The reflected two-point form ofthe Gibbs measure, at the two ENDS of a path, for REAL observables of a SINGLEslice, is non-negative under the parity and coupling hypotheses stated below-- at even separation for every beta, and at every separation exactly for beta>= 0 -- and the Gram matrix of a finite family of such observables is positivesemidefinite under the same hypotheses. THIS IS NOT YET THEOSTERWALDER-SCHRADER AXIOM, which quantifies over observables of the wholepast half-chain and over complex ones. The half-chain algebra, the reflectionmap and the sesquilinear form are not built here; the scope section says whatis missing and why it is a construction rather than a further inequality.Eleven papers in this lane end with REFLECTION POSITIVITY IS UNTOUCHED. Thistouches the endpoint form of it, and the interesting part is what the spatialsource weight does -- namely nothing. Earlier papers do prove statements aboutthe coupled kernel; specGap < lambda holds there too. What is new is that thisresult is UNCHANGED by the weight: same hypothesis, same conclusion, and noconstant that depends on w. Every earlier coupled-kernel statement contains anumber that moves when w does. Positivity is the SIGN of a quadratic form, andconjugation by sqrt(w) is a congruence, so there is nothing for the weight tomove.TWO REFLECTIONS, TWO HYPOTHESES. The reflected two-point sum is with v the dressed observable, and the two cases have genuinely differentcontent. Through a SITE (N even) the sum is a square, hence non-negative forEVERY beta -- negative coupling included -- with nothing used but symmetry ofthe kernel. Through a BOND (N odd) it is , which needs Kitself positive semidefinite; that holds exactly for beta >= 0, by aninduction on the extent.AND THE SECOND HYPOTHESIS IS ACTIVE. At beta < 0 and odd separation thesingle-site sign observable gives the reflected sum in closed form, 2(e^beta -e^-beta)^N, which is negative. Congruence is invertible, so the same witnessdivided by sqrt(w) shows the COUPLED kernel is indefinite below zero for EVERYstrictly positive weight, at every extent with at least one site. At L = 0there is nothing to witness -- one configuration, and there the decoupledkernel is 1 while the coupled one is the positive scalar w(empty) -- and thatexception is recorded rather than left to be found. The boundary beta = 0 issharp and exhibited, not inferred.WHAT THIS IS NOT. Neither the full axiom (above) nor any reconstruction: thephysical Hilbert space as the quotient of the past algebra by the null spaceof this form is not built. And the degeneration of specRatio(L) under a ringweight is MEASURED, not proved; no theorem here or elsewhere in the lane saysthe weight destroys uniformity, only that no uniform bound is proved for it.Nothing here concerns uniformity in the extent, SU(N), the continuum limit, orthe Yang-Mills mass gap.ON THE PRE-REGISTRATION. Three ACTIVE gates were committed before any Lean waswritten, and the gates section reports what happened to each. A fourth ispreserved there and not counted: the original Gate B, which failed because ofa design error of ours, and which two of the three active gates replaced.
Research paperHistorical importARR-2026-5SGNYDASRB9V9VF9 · v1
Lluis Eriksson
We formalize in Lean 4 the thermodynamic limit of bounded local Gibbs expectations for a periodic lattice gauge model in a uniform Kotecky-Preiss regime. The proof treats the complete finite-volume sequence: an exact one-volume marked expansion cancels the extensive far gas algebraically, common-window terms are transported exactly, and the remaining boundary contribution is bounded by an existing volume-uniform pinned cluster tail. The resulting explicit Cauchy modulus tends to zero, so completeness constructs an infinite-volume positive normalized real local state. On the intrinsic integer-coordinate local-observable algebra, the state carries a genuine additive action of Z^d and is invariant under every integer translation, including inverses. For SU(2), Haar probability measure, and the physical Wilson plaquette energy Re tr(U), the hypotheses are discharged throughout the explicit punctured intervals 0 < |beta| <= 10^-5 in d=2 and 0 < |beta| <= 10^-6 in d=4. We construct a genuine centered free-boundary exhaustion and prove that its complete cofinal sequence converges to the same state as periodic boundary conditions. The normalized finite-volume two-plaquette truncated-correlation bound also passes to the state under explicit eventual realization and separation hypotheses. We do not claim arbitrary boundary conditions, a C*-algebraic state, a continuum limit, Osterwalder-Schrader reconstruction, or progress on the continuum Yang-Mills mass-gap problem.
Research paperHistorical importARR-2026-3EXXMXJD119J1R6R · v1
Lluis Eriksson
Every rate in this lane so far has been a fixed-extent rate. The gap paper provedstrict spectral separation at each extent and said plainly that it was notuniform; the modulus paper gave that separation a number, specRatio(L), andreported measurements saying the number tends to 1 outside the disordered region.A geometric bound whose rate tends to 1 is empty in the volume limit, so nothingin the lane survived L -> infinity, and the word CLUSTERING was never used.PROVED. For the DECOUPLED kernel - the transfer kernel at constant source weight- the modulus is specRatio = tanh(beta) at EVERY extent, with L nowhere in it.Both directions: an operator bound by induction on the extent, and attainment bythe single-site observable the extent paper already built. Composing with themodulus paper's endpoint, the normalised Gibbs two-point function obeys|E[A(X_0)A(X_N)]| <= C_A tanh(beta)^N past one threshold serving everyobservable - a bound whose RATE contains no L, and therefore the first statementin this lane that survives the volume limit.WHY THE PROOF IS NOT THE SPECTRAL DECOMPOSITION. The decoupled kernel is aproduct over sites, so its spectrum is a product; that route needs the spectrumof a Kronecker power, which the library does not carry. It is not needed. Themodulus paper proved that specGap is the GREATEST norm ratio on the fluctuationsector, so bounding it above is an operator inequality and nothing else, and thatfalls to induction: the even part of an observable keeps its mean zero andinherits the rate, the odd part keeps nothing and gets only Schur's test, and thetwo recombine EXACTLY, because tanh(beta) Z = D with Z the row sum and D the oddeigenvalue of a single bond.NOT PROVED, AND A JUDGE THAT FAILED. The COUPLED kernel is untouched. Before anyof this was written we pre-registered two falsifiable predictions. The first -that the decoupled rate is exactly tanh(beta) at every extent - passed to 1e-16,and authorised the work above. The second - that the coupled uniformity boundaryis the Onsager curve - failed on one of eight pre-registered cells, and it staysfailed: that claim is reported as NOT ESTABLISHED, not softened. At constantsource weight the spatial slices are independent, so what is proved here is astatement about a product measure; that is exactly why it is reachable, and it issaid in the paper rather than left to be noticed. Reflection positivity isuntouched, and nothing in this paper is a claim about SU(N), the continuum limit,or the Yang-Mills mass gap.
Research paperHistorical importARR-2026-1A7WS4SDF0894BPS · v1
Lluis Eriksson
Two companion papers were left carrying the same debt from opposite sides. Thegap paper proved that every eigenvalue of the coupled slice other than thePerron eigenvalue is STRICTLY smaller in modulus, and said plainly that thisprovides NO MODULUS of separation. The bridge paper proved that the Gibbscorrelations of the spatial system are matrix elements of a self-adjointtransfer operator, and obtained geometric decay only UNDER A CONTRACTIONHYPOTHESIS IT DID NOT DISCHARGE. The missing step is identical in both: finitelymany strict inequalities are not an operator-norm bound.PROVED. We construct specGap, the largest |mu| over the eigenvalues differentfrom the Perron eigenvalue lambda, and prove specGap < lambda. We then prove theoperator bound: for every observable u orthogonal to the Perron vector,||Ku|| <= specGap*||u||. That is exactly the hypothesis the bridge papercarried, so its bound becomes unconditional. The bound is SHARP: whenever thestate space has at least two points, some nonzero fluctuation observable attainsit. The argument splits at specGap = 0, where the maximising index need notsupply a NON-PERRON eigenvector (it does supply the Perron one), hence none inthe fluctuation sector. Stated about an object too: the set of Rayleighnorm ratios on the fluctuation sector has a greatest element, equal to specGap(same two-point hypothesis: with fewer, that set is empty).A WARNING WE STATE BEFORE ANYONE ELSE HAS TO. specGap < lambda is NOTspecGap < 1: the kernel is unnormalised, so both are typically far above one andspecGap^N GROWS. On its own the unnormalised bound controls growth, it does notexhibit decay. The rate that is below one is the RELATIVE one,specRatio = specGap/lambda < 1, and the decay statement is that the fluctuationcontribution is suppressed by specRatio^N RELATIVE to the Perron scalelambda^N. Both forms are proved; only the second is called decay.The step that does not follow from the inequalities is the one abouteigenvectors AT lambda: geometric simplicity, proved in the Perron paper for anarbitrary eigenvector rather than a positive one, makes them INVISIBLE to afluctuation observable, so the top term of the spectral sum vanishes instead ofmerely being bounded.NOT PROVED. specGap DEPENDS ON THE EXTENT, and nothing here bounds it away fromlambda uniformly. Direct diagonalisation gives specGap/lambda = 0.9205, 0.9829,0.9964, 0.9992 at L = 2,3,4,5 for one parameter pair: a geometric bound whoserate tends to 1 is empty in the limit, and that is reported, not hidden. Thenormalised Gibbs EXPECTATION is now bounded too: splitting the dressed constantobservable along the Perron direction bounds the partition function below withno eigenbasis index identified, so at a fixed extent the two-point function isbounded by C*specRatio^N past an explicit threshold. That rate depends on theextent, so this is not clustering. The bound and its attainmentare both proved, which is what it means for specGap to be the operator norm onthe fluctuation sector; what is NOT done is introducing that norm as a definedobject and proving an equation about it. And the threshold N_0 does not dependon the observable: it is built from the dressed CONSTANT observable, so ONE N_0serves every fluctuation observable at once and only C sees A. Reflectionpositivity is untouched, and nothing in this paper is a claim about SU(N), thecontinuum limit, or the Yang-Mills mass gap.
Research paperHistorical importARR-2026-2TTA0WSQBT89X964 · v1
Lluis Eriksson
Four companion papers studied an OPERATOR: a strictly positive kernel on thespatial configuration space of a Z_2 slice, its Perron vacuum, and the strictseparation of its spectrum. None of them exhibited a MEASURE. That omission isthe kind this programme is built to notice: a transfer operator is a matrixuntil something says which Boltzmann weights it transfers, and until then avacuum is an eigenvector and a gap is a statement about eigenvalues. Neither isyet a statement about a statistical-mechanical system.PROVED. We define the two-dimensional Gibbs weight of the spatial system fromBoltzmann factors alone - a spatial factor at every time slice, a time-bondfactor between consecutive slices - and prove the DRESSING IDENTITY: that weightequals the path weight of the SYMMETRISED kernel multiplied by a factorsupported entirely on the two boundary slices. Hence every UNNORMALISED Gibbstwo-point sum of the spatial system is a matrix element of an iteratedSELF-ADJOINT transfer operator between boundary-dressed observables, and theNORMALISED expectation is the RATIO of two such matrix elements - the numeratoralone is not the correlation. The generic half holds for an arbitrary symmetrickernel on an arbitrary finite type; no positivity and no structure of theconfiguration space enter it. We further show that the operator the bridge landson is the one the companion papers analysed, that the fluctuation sector isinvariant, and that under an explicit contraction hypothesis the connectedtwo-point function decays geometrically in the time separation.NOT PROVED, AND THIS IS THE POINT OF THE LAST SECTION. The contractionhypothesis is carried as a theorem hypothesis and is NOT discharged. Thecompanion papers prove a STRICT gap with no modulus, and a strict inequalityamong finitely many eigenvalues does not by itself produce the operator-normbound a decay rate requires. Converting one into the other needs the spectralmaximum over the fluctuation sector, which is not constructed here. Inparticular NOTHING UNIFORM IN THE SPATIAL EXTENT is obtained, and none issuggested: with r = r(L) approaching 1, the bound is empty in the limit.Reflection positivity is not addressed. Nothing in this paper is a claim aboutSU(N), the continuum limit, or the Yang-Mills mass gap.
Research paperHistorical importARR-2026-1A608PTGTC9XX9JZ · v1
Lluis Eriksson
A companion paper supplied the vacuum of the coupled Z_2 slice at every spatialextent: a strictly positive eigenvector, unique up to scale, carrying thespectral radius. It listed PERIPHERAL SEPARATION as out of scope, and thatomission is not cosmetic - without it |mu| <= lambda leaves mu = -lambda open,and no gap follows at all.This paper closes it, and then draws the consequence that matters, which isnegative.PROVED. For a strictly positive kernel on a finite nonempty type, -lambda is notan eigenvalue, hence every real eigenvalue other than the Perron eigenvalue isSTRICTLY smaller in absolute value. Specialised to the coupled slice: at everyextent L, every beta, and every strictly positive source weight, the transferoperator has a strict spectral gap. The proof of peripheral separation avoidsthe equality case of the triangle inequality, which is where the classicalargument spends its effort: writing u = |w|, p = u - w and q = u + w, one getsA p = lambda q and A q = lambda p, so a nonzero p would make A p strictlypositive, hence q strictly positive, hence w nonnegative, hence p = 0.The separation is then extended from the real eigenvalues to ALL of them. Thecoupled kernel is conjugate by a positive diagonal to its symmetrised form,which is symmetric; and a real symmetric kernel has real eigenvalues, by acomputation that pairs the eigenvector against its image twice and is tworearrangements of a double sum. So there are no complex peripheral eigenvaluesleft to exclude, and the strict gap is a statement about the whole spectrum.That composition is itself a single machine-checked theorem(coupled_gap_all_eigenvalues), not a step left to the reader. We also deliverthe vacuum in Euclidean normalisation, norm(Omega) = 1 with T Omega = Omega.NOT PROVED, AND THIS IS THE TITLE. The gap is STRICT, not QUANTITATIVE: thetheorem provides no modulus of separation, and in particular nothing uniform inL. Direct numerical diagonalisation shows the subdominant ratio running0.9205, 0.9829, 0.9964, 0.9992 at L = 2,3,4,5 for beta = 0.8, gamma = 1.2 -collapsing towards 1. That computation is reported as measured and unproved, andno theorem here depends on it; its role is to say that a paper reporting onlythe positive half would be reporting the half that does not matter.Nothing in this paper is a claim about SU(N), the continuum limit, or theYang-Mills mass gap.
Research paperHistorical importARR-2026-2YDE42KFDY8Y3AA0 · v1
Lluis Eriksson
Two companion papers established, for a Z_2 lattice gauge slice with a spatialcoupling, that the elementary route to the vacuum stops, and that the naturalreplacement - the Hilbert projective metric - is blind to the coupling anddegenerates in the volume. Both had to work around the same absence: the pinnedmathlib carries no Perron-Frobenius theorem. The first paper could therefore onlysay the vacuum had become unavailable; the second had to build its dominationbound from scratch, and could exhibit the vacuum in closed form only at two sites.This paper discharges that dependency. For a strictly positive kernel on a finitenonempty type we prove, in Lean 4 with mathlib: a strictly positive eigenvectorEXISTS; its eigenvalue is strictly positive; any two strictly positiveeigenvectors are proportional and share their eigenvalue; and every realeigenvector for that eigenvalue is a scalar multiple of it. Together with thedomination theorem of the companion paper this gives the Perron statement thelane needs: the eigenvalue is the spectral radius.The existence proof does not use a fixed-point theorem, because the pinnedmathlib revision contains none. It maximises r over the compact set of pairs(r,x) with x in the simplex and r x <= A x; maximality forces equality, since astrict inequality anywhere would let one further application of A produce anadmissible pair with a larger r. The bound that keeps the set compact is obtainedby summing the constraint: r = r * sum x <= sum (A x).The application is the point. At EVERY spatial extent, and for EVERY strictlypositive weight on the source configuration - the class that contains the coupledkernel of the first paper - the vacuum exists, is unique up to scale, and carriesthe spectral radius. The obstruction of that paper was never an absence; it wasan unavailability, and it was an unavailability of one route rather than of theobject.NO SPECTRAL GAP IS PROVED HERE, uniform in the volume or otherwise, and none isclaimed. Nothing in this paper is a claim about SU(N), the continuum limit, orthe Yang-Mills mass gap.
Research paperHistorical importARR-2026-6M8PMVHTMH8YZ9ND · v1
Lluis Eriksson
A companion paper proved that when a spatial coupling is switched on in a Z_2lattice gauge slice, the transfer kernel loses constant row sums, so the uniformvector is no longer fixed and the elementary route to the vacuum stops. Thestandard replacement, when row sums fail, is the Hilbert projective metric: astrictly positive kernel contracts it, and the contraction factor bounds thesubdominant spectral ratio. This paper asks what that replacement gives here, andanswers in Lean 4 with mathlib.It gives no coupling-sensitive and no volume-uniform information, for twoindependent reasons, and both are proved. First, BLINDNESS: the projectivecross-ratio is invariant under multiplication by any nowhere-zero function of thesource configuration alone. The coupled kernel is exactly such a product, so themetric assigns the interacting and the non-interacting kernels the same diameterat every spatial extent - the route cannot see the coupling at all. Second,VOLUME DEGENERATION: two constant configurations realise the cross-ratioe^(4 beta L), so every admissible projective diameter is at least 4 beta L andevery contraction factor obtainable this way is at least tanh(beta L), which lieswithin 2 e^(-2 beta L) of the trivial bound 1. At the one place where the truthis known - the decoupled kernel, whose subdominant ratio the companion papercomputes to be exactly tanh beta at every L - this route already returnstanh(beta L) instead. The degeneration is the method's, not the model's.We then hand over the object the elementary route stopped producing, at thesmallest interacting size. In the character basis the coupled two-site kernelsplits into two 2x2 blocks, and we exhibit a strictly positive eigenvector inclosed form, together with a second exact eigenpair. The identity A - B = 4between the two decoupled even-sector eigenvalues drives every estimate. Theblindness is proved two-sided, so it covers the symmetrised conventionw^(1/2) K w^(1/2) as well, and the positive eigenvector is proved to dominateevery eigenvalue, real or complex - so its eigenvalue is the spectral radius,which is the Perron statement this development needs and proves without aPerron-Frobenius theorem in the library.NO VOLUME-UNIFORM STATEMENT ABOUT AN INTERACTING SYSTEM IS PROVED HERE, AND NONEIS CLAIMED; the general-L behaviour is recorded separately as measured andunproved. Nothing in this paper is a claim about SU(N), the continuum limit, orthe Yang-Mills mass gap.
Research paperHistorical importARR-2026-3CEWH763KG84B8TM · v1
Lluis Eriksson
For beta > 0 let I_m = I_m(beta) denote modified Bessel functions ofthe first kind, and set a_m = I_m^2 [(m-1) I_(m-1)^2 + (m+1) I_(m+1)^2], b_m = m I_m^4, F_A(t) = sum_(m>=1) a_m sin(mt), F_B(t) = sum_(m>=1) b_m sin(mt), E(t) = F_A(t)/(2 F_B(t)).The global ratio-monotonicity problem for the surface expansion of atwo-dimensional SU(2) lattice gauge observable is: (i) F_B > 0 on(0,pi), and (ii) E' < 0 on (0,pi), for every beta > 0. Bothstatements are proved. Positivity of F_B has two exact proofs. Ratiomonotonicity is reduced to exact algebraic identities andoutward-rounded interval certificates: small and compact beta arehandled by pair identities and interval Taylor models;20 <= beta <= 1000/9 by a direct Wronskian cover; andbeta >= 1000/9 has three certified moving-edge lambda lanes. Theremaining lambda >= 3 lane is closed by the exact identityE'/(-sin(t/2)/2) = Q + X_full, where Q > 19/20, an exactmain--mirror--rest decomposition, and a division-free covariancecertificate proving X_main > -1/20 on two adjacent rectangles thatcover the full angular interval. The exact near and far relay marginsare positive. All load-bearing production and independent replaytranscripts are checked for exact rational coverage, dependencyhashes, strict outward-rounded decision endpoints, and byte equality.The structural core is exact: E is, as an algebraic identity, the meanof cos(psi) under the midpoint law of a four-step killed von Misesbridge; the generating kernels reduce, via the Neumann additiontheorem, to two-dimensional integrals of a single Bessel functionwhose saddle deficit is an exact sum of two squares; and exact saddlecancellations yield the coefficients of the verified closedsecond-order law E = cos(t/2)(1 - c(t)/beta) + O(beta^-2), c(t) = (4 cos^2(t/4) - 1)/(2 cos(t/4) cos(t/2)).Three certified negative results (interval arithmetic, twoimplementations, nested enclosures) kill every monotone full-pathcoupling, with an exact mechanism at threshold beta |cos t| = 3/2. Atthe pi endpoint we also give exact identities for the cubiccoefficient c_3 (telescoped alternating form, integral form, parity)together with its verified prefactor law. Every claim is labelledexact / certified / verified; the machine-checked lemmas are Lean4/Mathlib, machine-checked modulo classical Bessel inputs carried asnamed hypotheses.
