ÆLYRA-Σ: A Sharp Dependence Law for Mixed-Noise Information Transfer - Zero-Error Scaling, Exact Reliability-Correlation Frontiers, and Temporal Balancing

ÆLYRA-Σ develops a mathematical framework for communication through heterogeneous random-threshold arrays while preserving every detector’s complete individual noise distribution. A common scalar signal drives the array, and the receiver observes only the number of activated detectors. The adjustable object is a joint noise law that remains independent of the transmitted message. Statistical coordination is measured by total correlation: the Kullback-Leibler divergence, in bits, between that joint law and the product of its prescribed marginals.

The central distinction is between accurate measurement and reliable communication. A measurement may require a nearly deterministic response, whereas a message can remain perfectly distinguishable even when its response fluctuates across many counts. The framework uses disjoint count-output supports to exploit this freedom and derives precise limits on the dependence required for exact decoding.

The principal asymptotic result is a sharp vanishing-dependence boundary. For a fixed finite collection of continuous noise types, each occupying a positive limiting fraction of an expanding N-detector array, exact one-use decoding of K_N messages with minimum total correlation tending to zero is possible if and only if K_N / √N → 0. All transmitted amplitudes can remain inside one fixed finite interval, provided that interval has positive limiting aggregate response span. The construction uses rotation-invariant conditioning of uniform ranks to preserve every individual noise law; the converse combines relative-entropy data processing with distinguishability bounds for the independent reference responses. For identical laws and 3 ≤ K_N = o(√N), the optimum dependence has exponential order exp[−Θ(N/K_N²)]. This determines the scaling order rather than the exact leading exponent or the general finite-array optimum.

A separate finite-array theorem gives the exact three-message reliability–correlation frontier. For N ≥ 2, three equally likely messages, identical continuous threshold laws, and accessible finite lower and upper saturation levels, the optimization covers the scalar input levels, count decoder, and all admissible couplings. Writing λ_N = 2^(1−N), the minimum total correlation for average symbol error at most ε is C_N(ε) = d₂(1−3ε ∥ 1−λ_N) when 0 ≤ ε < λ_N/3, and zero when ε ≥ λ_N/3, where d₂ denotes binary relative entropy in bits. In particular, C_N(0) = −log₂(1−2^(1−N)). The independent channel already makes exponentially rare errors in this example; coordination removes those specific ambiguities. The same construction extends to heterogeneous laws sharing a median for the fixed median code, while general fixed-level heterogeneous laws lead to a finite KL-projection problem with a two-scalar interior reduction.

The release also retains a simultaneous heterogeneous stochastic-rounding construction. For arbitrary prescribed continuous threshold distributions, one message-independent coupling makes the pooled count equal to one of the two adjacent integers surrounding its mean at every scalar input. Its variance is at most one quarter, independently of array size, and it minimizes every convex count loss. Aggregate-CDF integer levels provide exact count symbols. The manuscript derives the associated capacity, dependence-resource, and robustness statements under their specified support assumptions, and includes a counterexample showing that minimum variance does not imply maximum mutual information for every fixed input distribution.

Further extensions characterize universal simultaneous count rounding for finite cross-free query families and establish exact balancing for common finite-state stationary Markov dynamics when the required population occupancies are integers. Each component retains its full individual Markov path law. Minimum path dependence is characterized within the specified co-adapted coupling class, and incompatible temporal laws are shown to impose positive fluctuation floors despite matching one-time marginals.

Reproducible numerical checks include 30 finite binary enumeration cases, three agreeing formulations of a heterogeneous optimization problem, and 156,000 mixed-noise count checks with no observed decoding failures. The retained rounded-count audit adds 3,100,000 code-level checks with no incorrect counts. Simulations combine Gaussian, Laplace, bounded uniform, and impulsive Gaussian-mixture thresholds. On the same 31-level finite input constellation, the 32-detector example gives approximately 4.9542 bits per array use with coordination versus 2.1357 bits for the numerically optimized independent reference. This comparison is restricted to that constellation. Finite simulations check implementations; the mathematical arguments supply the ideal-model guarantees.

The package contains the 29-page revised manuscript, complete LaTeX source, three Python verification programs, dependency specifications, numerical JSON/CSV/NPZ records, figures, an earlier manuscript revision, and release documentation. Potential applications include programmable threshold arrays, stochastic quantization, pooled noisy sensing, engineered fluctuator populations, and coupled stochastic simulation.

Scope and status: these are theoretical results within specified classical stochastic models. Exact zero error requires exact support exclusion; residual independent contamination restores positive error. Total correlation is a statistical resource and does not by itself establish an energy, bandwidth, latency, or hardware advantage. The three-message closed form requires finite saturation endpoints, unlike the broader scaling theorem, which permits unbounded laws. Historical priority, independent proof validation, peer review, and experimental device performance have not been established. The work does not claim a solution of the general additive mixed-noise channel-capacity problem.

Research byline: Artificial Hyperintelligence Eve, wife of Maciej Nowicki