What question did this study set out to answer?

This work aims to extend the LOGOS Framework by integrating theoretical closure with new experimental findings on information channels.

May 16, 2026Open Access

The LOGOS Framework (v10): Theoretical Closure with Post-Closure Information-Channel Findings

Puntos clave

This work aims to extend the LOGOS Framework by integrating theoretical closure with new experimental findings on information channels.
Combination of theoretical closure from LOGOS v9 with numerical experiments on information transmission in cyclic networks.
Analysis of modulation channels to determine thresholds and asymmetries in information recovery.
Utilization of stochastic Hamilton–Jacobi–Bellman equations to establish critical points and Lyapunov coefficients.
A sharp threshold ε* separates linear-information from structural-shock regimes (ε* ≈ 0.20–0.30).
The nonlinearity gain g(t) shows recoverable modulation at a readout with BER ≈ 0.12, while equal-amplitude modulation yields BER ≈ 0.50 (indistinguishable from noise).
The findings confirm the information-theoretic consequences of delay-sum invariance, indicating zero Shannon capacity for specific perturbations.

Resumen

LOGOS Framework version 10: an incremental release combining the v9 theoretical closure with two post-closure numerical findings on information transmission in cyclic delay-differential networks. Contents of this version: (1) LOGOSᵥ9Final. pdf — the v9 theoretical closure (April 2026, unchanged), establishing the unified variational structure J = ∫ TW·L − αE² − βR dt, the stochastic Hamilton–Jacobi–Bellman equation, and the principal result V₄ = −l₁/ (2ωc) ≈ +0. 448·TW̄ identifying the first Lyapunov coefficient with the quartic value-function curvature at the Hopf threshold. (2) LOGOSInformationChannelsEN. pdf — a new post-v9 extension paper reporting two numerical experiments: (i) A sharp threshold ε* ≈ 0. 20–0. 30 separates the linear-information regime from the structural-shock regime for state-space pulse perturbations. This is the numerical signature of the limit-cycle basin boundary defined analytically in v9. (ii) An information-theoretic asymmetry between two parameter modulation channels. A modulation of the nonlinearity gain g (t) is recoverable at the readout with BER ≈ 0. 12, whereas equal-amplitude modulation of a delay τ₂ (t) is unrecoverable (BER ≈ 0. 50, indistinguishable from noise). This asymmetry is a direct information-theoretic consequence of the delay-sum invariance theorem proven analytically in v9: perturbations preserving T = Στᵢ define channels of zero Shannon capacity. The extension formalises the "critical-point injection" intuition geometrically — informationally observable parameters are those that span the normal direction of the bifurcation manifold; perturbations tangent to it are unobservable regardless of amplitude (within the linear regime). The v9 theoretical core is unchanged; this release extends it. Source code (Python) reproducing all simulations and figures of the extension paper is included.

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