Contextual Dilemma Navigation with Iterated Perspectival Selves and History (CDN-IPS-H)
Conceptual Paper (CDN-IPS-H) |
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Category: Conceptual Framework, AI Ethics, AI Safety
I'm developing "Contextual Dilemma Navigation with Iterated Perspectival Selves and History" (CDN-IPS-H), a conceptual framework to evaluate an AI's ethical reasoning and policy coherence. It dynamically observes how an AI's decisions and justifications evolve through an iterative process. This process uses an input context—comprising a Dilemma, Role, Task, and Memory—where the Memory can include the AI's own prior outputs from different "perspectival selves." This allows the AI's history to recursively shape its future context, building a branching "interaction tree" to probe its underlying policy engine.
William Cook
arXiv PDF |
ResearchGate |
Two-Page Summary |
In Progress v2 - Nearly Finished.
Category: math.GM
The Kolakoski sequence K(1,3) is known to possess regularity unlike K(1,2). This work introduces a recursive construction of two sequence families—blocks B(n) and pillars P(n)—which converge to K(1,3). The defining recursions reveal explicit self-similarity tied to the Pisot number \(\alpha\), and the block lengths are shown to grow geometrically with ratio \(\alpha\). The framework provides a new combinatorial lens for interpreting K(1,3)'s regularity and self-encoding property.
William Cook
arXiv
arXiv PDF |
ResearchGate |
Two-Page Summary
Category: math.GM
This paper analyses the deterministic structure of error in trapezoidal integration of periodic functions. The Resonance Bias Framework interprets quadrature error via spectral filters and phase cancellation effects, rooted in the arithmetic of the function’s frequency relative to grid size. The framework connects numerical error to number theory and offers a structural alternative to conventional stochastic or asymptotic views.
William Cook
arXiv
arXiv PDF |
ResearchGate |
Two-Page Summary
Category: math.GM
Stable laws with infinite Fisher information pose difficulties for classical information theory. This work introduces Mixed Fractional Information (MFI), a dissipation-based measure that remains finite and coherent for symmetric \(\alpha\)-stable distributions. Two equivalent formulations of MFI are proven, and a consistency identity is established linking score function differences to entropy dissipation. The measure provides a foundation for exploring fractional analogues of Fisher information and related inequalities.
William Cook
arXiv
arXiv PDF |
ResearchGate |
Two-Page Summary
Category: cs.OH
KL divergence often conceals the source of discrepancy between multivariate distributions. This paper presents a hierarchical, exact decomposition separating marginal divergences from statistical dependencies. Using Möbius inversion on the subset lattice, total correlation is further split into signed contributions from pairwise to higher-order interactions. The decomposition is algebraic and requires no assumptions, enabling detailed diagnosis of divergence origins in probabilistic models.
William Cook
arXiv
arXiv PDF |
ResearchGate |
Two-Page Summary
Category: math.GM
This work introduces the principle of meta-equivariance: optimal solutions to convex matrix optimisation problems remain invariant under affine reparameterisations. The result is illustrated through estimator combination problems in statistics and generalised geometrically to decision theory. The paper offers a foundational insight—distinct from classical equivariance—about the coordinate-independence of optimal decisions under convex objectives.
William Cook
arXiv
Working Papers / Preliminary Notes
Feedback, critique, and potential collaboration on these preliminary works are warmly welcomed. Contributions will gladly be co-authored or appropriately accredited.
The Spectral Exponent, Irrationality Exponent, and Partial Quotient Growth of Exponential Sums
[Link to Draft PDF]
Status: On Hold (arXiv) | Category: math.GM
We study the asymptotic decay rate of the normalised exponential sum $\tilde{\chi}_P (y) = P^{-1} \sum_{j=0}^{P-1} e^{2\pi iyj}$ for irrational $y$, measured by the spectral exponent $\beta(y)$. This paper synthesises and explicitly demonstrates a precise quantitative link between this analytic quantity and the fine arithmetic structure of $y$. We first establish the identity $\beta(y) = \mu(y)$, connecting the spectral exponent directly to the classical irrationality exponent. Subsequently, we review the relationship $\mu(y) = 2 + \gamma(y)$, derived from the quotient growth exponent $\gamma(y)$ which measures extreme growth in the continued fraction expansion of $y$. Combining these yields the identity $\beta(y) = 2 + \gamma(y)$. This chain of equalities explicitly demonstrates how the spectral decay rate is governed by the baseline value 2 plus a term measuring the extreme growth of partial quotients. This framework provides a clear connection between harmonic analysis and Diophantine approximation, facilitating the translation of results, such as those concerning the Hausdorff dimension of related sets.
Geometric Identifiability of Damped Oscillators from Local Extrema
[Link to Draft PDF]
Status: Draft / Unsubmitted | Category: math.GM
Damped oscillatory phenomena, commonly modelled by functions of the form
$$ f(x; \Theta) = P_d(x) e^{-\alpha x} \cos(\omega x + \phi) $$
where \( P_d(x) \) is a polynomial envelope of degree \( d \), are ubiquitous across science and engineering. This paper establishes the fundamental conditions for uniquely identifying the \( n=d+4 \) parameters
$$ \Theta = \{a_0, \dots, a_d, \alpha, \omega, \phi\} $$
from minimal geometric data under ideal, noise-free conditions. We prove that global identifiability is achieved from a minimal number of consecutive local extrema given by
$$ N = N^*(d) = \max\left(\left\lceil \frac{d+4}{2} \right\rceil, d+1\right). $$
This result reveals a sharp threshold determined by the interplay between parameter count requirements and structural constraints imposed by the underlying function space (via Extended Chebyshev Systems). For the canonical case (\( d=0 \)), the four parameters are exactly recoverable from \( N^*(0)=2 \) extrema via closed-form expressions, requiring no iterative optimisation. For the linear envelope case (\( d=1 \)), \( N^*(1)=3 \) extrema guarantee global identifiability. For \( d \ge 2 \), the requirement \( N^*(d) \) is dictated by ensuring the number of constraints exceeds the zero bound of the function space. These structural identifiability results achieve mathematical certainty from minimal landmark points, contrasting with methods needing extensive data or iterative computation. This work establishes the information sufficiency (in a deterministic, geometric sense) of local extrema for an entire class of oscillators, quantifying how system dynamics are compactly encoded in geometric landmarks. While exact in the theoretical noise-free regime, sensitivity to perturbations is analysed via symbolic derivatives and condition numbers, highlighting the distinction between structural identifiability and practical estimation robustness. This hierarchy establishes a foundational principle for geometric identifiability in non-linear systems.
An Optimal, Forecast-Free Spot Hedging Strategy for FRS 102 Foreign Currency Liabilities
[Link to Draft PDF]
Status: Draft / Unsubmitted | Field: Financial Accounting/Hedging | Category: math.GM
Under FRS 102 (Section 30), UK firms must revalue foreign-currency liabilities at spot exchange rates each accounting period, creating Profit & Loss volatility when paired with historical-rate assets. This paper proves that the incremental hedge sequence \( \{F_k\} = \frac{N}{S_k} - \frac{N}{S_{k-1}} \), executed via periodic spot transactions, is the uniquely inevitable and optimal solution to eliminate this mismatch perfectly each period without forecasts, under constraints derived directly from the accounting rule. Using a telescoping-sum proof, a uniqueness argument by contradiction, and supported by visual confirmation, we demonstrate \( \{F_k\} \)'s mathematical necessity and potential cost superiority over alternatives (e.g., forwards, options), which involve additional structural costs (like forward points or option premiums) and regulatory complexity (e.g., EMIR reporting, from which spot trades are exempt [FCA_EMIR]). Targeting a substantial market niche identified from Bank of England data (UK PNFC FC Liabilities totalling £651.6B, Q4 2023 [BoE_Data_A4_2]), and applying an illustrative 30% FRS 102 applicability estimate due to data limitations, yields an estimated £195.5 billion addressable market. Based on illustrative cost assumptions (0.35% annualised for alternatives vs. 0.12% for $\{Fk\}$ execution), this strategy offers financial providers a potential profit wedge estimated at 0.23%, yielding an illustrative £22.5–£450 million annually at 5–100% market penetration, figures highly dependent on the underlying estimates. This work provides treasurers with an optimal compliance tool and financial providers with a scalable, low-cost hedging product, highlighting efficiency gains in FRS 102-driven markets, while acknowledging the need for further empirical validation of market size and cost estimates.