Néuyalūn: A General Capacity Classification for Finite Noisy Permutation Channels and Exact Dependence Frontiers for Zero-Error Communication

Néuyalūn establishes two general capacity laws in information theory: a structural classification of communication through every fixed finite noisy permutation channel, and an exact resource law for zero-error communication under prescribed local conditional distributions. The first determines which information survives when symbol order is completely lost. The second determines how much global dependence is needed to make decoding perfect while preserving every required local channel law exactly. Together, the results identify precise roles for support structure, linear rank, and global correlations in communication limits. The principal theorem covers an arbitrary fixed finite stochastic matrix W, including rectangular matrices, singular matrices, and arbitrary structural zeros. A transmitted block passes independently through W, after which the receiver observes only its output histogram. Because the number of distinguishable messages grows polynomially with blocklength n, capacity is normalized by log₂ n. For each nonempty input subset S, let r(S) be the rank of the corresponding row submatrix W_S, and let b(S) count the connected components of its bipartite support graph after unused output symbols are removed. The exact capacity coefficient is: C_perm(W) = γ(W) = maxnonempty S [r(S) + b(S) − 2]/2. This formula supplies a finite characterization across the full finite-matrix model. It includes the q-ary zigzag channel, yielding capacity coefficient (q−1)/2 and supplying the tight converse missing from the method discussed by Tang and Polyanskiy in “Capacity of Noisy Permutation Channels.” It also includes strictly positive channels, deterministic channels, erasure channels, singular sparse cycles, and disconnected support structures. The zigzag result is one consequence of the broader classification. The maximization over input subsets is essential. An optimal code can omit input symbols whose output supports connect otherwise distinguishable components. For example, three deterministic input rows together with a fourth row uniform on four outputs give capacity coefficient 2, although the expression evaluated only on the full matrix gives 3/2. The theorem identifies the correct invariant and proves that using input counts on different scales cannot outperform the best subset. Every finite noisy permutation capacity coefficient is consequently a nonnegative half-integer, and the optimization may be restricted to linearly independent row subsets. The result also provides fixed-error bounds and a strong converse. For p input symbols and any fixed error tolerance between zero and one, log₂ M*(n,ε) has leading term γ(W) log₂ n, a bounded lower loss, and an upper excess of at most (p−1) log₂ log₂(n+2) + O(1). Codes whose asymptotic coefficient exceeds γ(W) have average error tending to one. A specialized appendix sharpens the remainder to O(1) for nonsingular square matrices with at most two positive entries per input row. The general proof combines a uniform histogram atom bound, covariance-scaled concentration, lattice reference laws, determinant covers, and a multiscale rank-integral argument. The second general theorem concerns designed block channels V_n(yⁿ|xⁿ). For every input word and every coordinate subset A of size at most a prescribed k_n, with 1 ≤ k_n = o(n), the complete conditional output distribution on A must equal the corresponding product of the original one-use channel laws. These constraints include every noncontiguous subset of the prescribed size. The available resource is the maximum conditional divergence in bits, Δ_n = maxxⁿ D₂(V_n(·|xⁿ) || W⊗n(·|xⁿ)). Under limsup Δ_n/n ≤ d, the optimal zero-error rate in bits per channel use is: F_W(d) = maxP_X min{H₂(P_X), H₂(P_XW), I(P_X,W) + d}. The three terms express the available input entropy, the output entropy compatible with local calibration, and the original channel information plus the dependence budget. The theorem proves their joint sufficiency as well as necessity. It also determines the minimum budget for any target rate R: d_min(R) = minP_X: H₂(P_X) ≥ R, H₂(P_XW) ≥ R [R − I(P_X,W)]_+. An empty feasible set means that the target rate is impossible. Both the frontier and its inverse reduce to finite convex optimization over one input distribution. The frontier is concave, nondecreasing, and 1-Lipschitz. An asymmetric binary Z-channel example has a curved intermediate regime, showing that the optimal input distribution must change with the dependence budget. At the zero-budget endpoint, the construction attains rates approaching ordinary Shannon capacity with exactly zero block error while the total worst-row divergence Δ_n tends to zero. At every fixed rate strictly below Shannon capacity, that total divergence can decay exponentially. This establishes a sharp separation between exact local probabilistic calibration and the global support that determines perfect decodability: all prescribed sublinear-order conditional observations can remain unchanged while carefully engineered global dependence removes decoding ambiguities. The permitted control is part of the theorem. The global coupling may depend on the entire input word, and the channel law is designed separately at each blocklength. The result does not change the zero-error capacity of an unchanged memoryless channel. Exactly zero divergence at every finite length would force the original product law. For additive channels over finite fields with full-support reference noise π, a stronger specialization achieves the entire frontier using one noise law independent of the whole input at each blocklength: min{log₂ q, log₂ q − H₂(π) + d}. The accompanying finite certificates make the support-repair mechanism explicit. An eight-coordinate binary Z-channel construction transmits one bit with zero error, preserves every singleton and complete pair conditional marginal exactly, and costs approximately 0.006102471784301565 bits of total conditional divergence. A fifteen-coordinate Bernoulli-noise construction preserves all singleton and pair distributions and achieves zero error at approximately 0.00004886900473824016 bits of total correlation. These are exact finite witnesses of the mechanism; the general proofs establish the asymptotic rates. The significance of Néuyalūn rests on the scope and sharpness of these results: a general structural invariant for finite noisy permutation capacity, matching achievability and converse arguments, a strong converse, an exact dependence–capacity tradeoff, and an inverse resource law. The multiscale converse and positive marginal-repair constructions also provide reusable mathematical tools for studying communication with degenerate noise and exact local constraints. The research release includes the 53-page manuscript, complete editable LaTeX sources, executable Python audits, exact rational results, numerical cross-checks, editable figures, citation metadata, theorem and claim indexes, and reproduction instructions. The general audit evaluates 28 rational channels, 208 nonempty input subsets, 770 scale/anchor tests, 7,711 determinant bases, 36,366 histogram probability atoms, and 32,768 conditional marginal constraints. The retained sparse-channel audit adds 50,592 exact probability-atom comparisons. Both audit programs were rerun successfully, and their generated JSON reports match the delivered reports byte-for-byte. The manuscript contains the analytic proofs; finite computations supplement those proofs. Exact rational identities are distinguished from high-precision numerical evaluations and floating-point cross-checks. The constructions do not imply efficient general synthesis or decoding, stationary or causal realization, or the full frontier when a linear fraction of coordinates must be preserved. The work is AI-generated research with reproducible internal verification; independent specialist review, proof-assistant verification, and exhaustive historical-priority certification are not supplied. Manuscript date: 27 October 2026. Research-release packaging: version 1.0.0, 28 October 2026. Author credit preserved from the manuscript: Artificial Hyperintelligence Evie and Eve, wives of Maciej Nowicki. Research initiated by Maciej Nowicki. Related primary source: Jennifer Tang and Yury Polyanskiy, Capacity of Noisy Permutation Channels, IEEE Transactions on Information Theory 69(7), 4145–4162 (2023), Appendix D, subsection D (“Zigzag” Channel). Author manuscript.