Conjecture 43: Norm-Dependent Collapse Threshold

This paper establishes a quantitative bridge between two pillars of the Syntony corpus: Conjecture 41 (optimal norm selection via tail-ratio) and the Syntonic Capacity Theorem (universal collapse threshold λ_c = ln(2) / (2 T_env)). The central question is whether the norm choice p used by an adaptive system to estimate its scale parameter affects its collapse threshold. Under an additive error model for scale estimation, the paper derives an explicit closed-form expression for the norm-dependent effective threshold: λ_c^eff = ln(2 + δ²(p) / σ²) / (2 T_env) where δ²(p) is the estimation variance of the scale parameter under norm p, given by the Cramér-Rao bound scaled by the asymptotic relative efficiency of the L^p estimator for the innovation distribution. The threshold is always greater than or equal to the ideal λ_c from the Syntonic Capacity Theorem, with equality achieved only in the limit of perfect estimation (δ² → 0). Quantitative predictions. For Laplacian innovations with N samples, the paper predicts an explicit gain of approximately 6.7% in the collapse threshold when using the optimal L¹ estimator (matched to the distribution's tail) compared to the naive L² choice. For Gaussian innovations the two choices coincide exactly, as expected. The formula extends continuously through the Lp family and connects to the tail-ratio ρ = σ / MAD introduced in Conjecture 41. Status. The result is explicitly framed as a conjecture rather than a theorem. The derivation relies on an additive error modeling assumption that is stated as a modeling choice rather than proved from first principles. The three mathematical steps of the chain 41 to 42b to 43 to 41 are established independently; closing the chain requires proving that the efficiency of τ-hat inherits the efficiency of matched L^β scale estimators via the delta method, which is stated as an open step. Testable predictions. The paper formulates three empirical tests. First, physical flight data from the SyntonicQuad drone platform, which experiences non-stationary non-Gaussian motor and turbulence noise, should show measurable improvement in tracking coherence when using C41-based adaptive norm selection. Second, deep learning training dynamics under heavy-tailed gradient distributions should show measurable improvement in convergence stability when using L¹-matched optimization. Third, financial time series under regime-switching volatility should show measurable improvement in out-of-sample tracking when using tail-adaptive norm selection. Relationship to the Syntonic Capacity Theorem. The Syntonic Capacity Theorem establishes that no strategy can beat λ_c for a system with perfect knowledge of σ². Conjecture 43 extends this to the realistic case where σ² must itself be estimated, and shows that the collapse threshold is then shifted by the estimation error, which depends on the norm chosen. This closes a loop: the ideal bound is never actually attainable in practice, and the effective bound depends on how well the system's measurement geometry matches the innovation distribution's tail. Scope. Like the Syntonic Capacity Theorem, the result is scalar, continuous-time, with i.i.d. innovations at discrete intervals. The additive error model is stated explicitly as a modeling assumption, not proved. Part of the Syntony corpus. This paper is a theoretical extension of the Syntony Principle (Canonical Edition V4.1, DOI: 10.5281/zenodo.17254395) and depends directly on the Syntonic Capacity Theorem (DOI: 10.5281/zenodo.19637032) and Conjecture 41 (DOI: 10.5281/zenodo.18911127). 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