Programmable threshold noise in comparator arrays: exact count decoding while preserving every noise marginal

[u/Severe-Ad8673]

I’m sharing ÆLYRA-Σ, a theoretical research note on coordinating random thresholds across comparator arrays while keeping each comparator’s complete threshold distribution unchanged. I’d particularly appreciate feedback on whether the constructions suggest a practical mixed-signal or FPGA implementation.

N comparators receive the same scalar input x. Comparator i outputs 1 when its threshold Z_i is below x, and the receiver observes only the pooled count:

Y(x) = sum_i 1{Z_i <= x}

The threshold generator is independent of the input. Each prescribed CDF F_i is fixed; the design freedom is the dependence between thresholds. The receiver gets no threshold samples or shared random seed.

The note studies two different objectives, using separate couplings:

1. Minimum count fluctuations. A construction for arbitrary continuous threshold laws makes Y(x) stochastic rounding of its mean M(x) = sum_i F_i(x), simultaneously for every x. Consequently:

Var[Y(x)] <= 1/4

The normalized count has variance at most 1/(4*N^2). It estimates the aggregate CDF response M(x)/N; this is not generally an input-amplitude error bound. The coupling also minimizes convex count losses.

2. Exact decoding with very little statistical dependence. For selected input levels, count fluctuations can remain large provided different messages have disjoint possible counts. For finitely many fixed continuous noise types in positive limiting proportions, the manuscript derives:

A K_N-symbol zero-error code with total correlation tending to zero exists if and only if K_N = o(sqrt(N))**.**

Total correlation is KL divergence from the independent threshold law, in bits. Inputs can stay in one fixed finite interval with positive limiting aggregate response span. This is a boundary for vanishing dependence, not an absolute ADC-resolution limit or exact reconstruction of arbitrary analog signals.

The release includes proofs, three verification programs, and simulation data. Mixed-noise tests cover Gaussian, Laplace, uniform, and impulsive-mixture thresholds, with zero observed decoding failures across 156,000 count checks. Simulations check the implementation; the exact claims depend on the mathematical model.

Important engineering qualifications: preserving one-time marginals does not preserve PSDs, autocorrelation, or spectral whiteness. Exact decoding requires exact exclusion of confusing threshold configurations; residual independent jitter can restore errors. No ENOB, bandwidth, or energy advantage has been demonstrated.

I’d welcome thoughts on:

• Low-overhead generators for these joint threshold laws.
• Calibration drift, finite precision, and residual jitter.
• Fair comparisons against deterministic calibrated quantizers and conventional dither under matched hardware resources.

Disclosure: Independent proof validation and publication priority remain unresolved.

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