Conjecture 41: Optimal Norm Selection in Adaptive Systems via the Tail-Ratio Observable

The Syntony law remains structurally unchanged; the adaptation occurs at the level of the statistical geometry used to estimate the scale quantities entering the law. Document version 1.1 (archived in Zenodo record version 2). The Syntony Principle states that the optimal temporal integration constant satisfies τ* = κ·γσ/γλ, where γσ and γλ denote scale estimators of dispersion and innovation. This note asks whether, under non-Gaussian noise, the norm used to estimate these scales can itself be selected from the same observables. For the symmetric Generalized Gaussian family, we show that the tail-ratio observable ρ = σ/MAD is a scale-free, strictly monotone function of the shape parameter β, with exact anchors √(π/2) for the Gaussian case and √2 for the Laplacian case. This provides a direct observable of distributional geometry from quantities already available in the adaptive law. Motivated by maximum-likelihood and M-estimator geometry, we then formulate Conjecture 41 in scale-estimation form: the Syntony law itself remains unchanged, but when the local noise follows a Generalized Gaussian law of shape β, the asymptotically efficient geometry for estimating the scale quantities entering the law is the matched norm L^β. Consequently, the induced estimate of τ* is conjectured to be most precise when p* = β. On the engineering side, the note introduces a continuous, parameter-free L2/L1 firmware interpolation driven by the tail ratio, exact at both endpoints and continuous between them. Proven results, conjectural steps, and firmware approximations are stated separately. The central open problem is to prove that temporal scale adaptation inherits, to first order, the same efficiency structure as the underlying scale estimators under Generalized Gaussian noise. This record contains the validation document (PDF). The theoretical framework is described in: Bronsard, J.-P. (2024–2025). The Syntony Principle: A Unified Geometric Framework for Adaptive Intelligence. Authorship and collaboration statement. All prose in this corpus was drafted by large language models under the author's direction and revised in author-directed loops. The author set the questions, chose the hypotheses, designed and ran every experiment, decided every retraction, and designed the corpus as a system (immutable hub-and-spoke record, concept and version identifiers, errata register, hash-frozen firmware, pre-registration gates). Where model-generated reasoning text survives in a published document, it is an artifact of this method and is recorded in the Corpus Status Register. The author alone approves every statement and is accountable for it; no language model is an author. RETURN TO HUB : https://doi.org/10.5281/zenodo.17254395