Thursday, August 27, 2026

The Adjustable Aperture: Algorithms as Mathematical Lenses on Information

John Swygert

August 27, 2026

Abstract

This paper develops the metaphor of the algorithm as an adjustable mathematical lens. A dataset can remain invariant while different transformations, coordinate systems, radices, objective functions, feature selections, and search procedures make different relationships easier or harder to discover. The idea emerged from a sequence of radix-fitness experiments in which stronger planet-specific claims attenuated under independent and metric-neutral tests, while a narrower observer-centered result survived: representation can affect discoverability for particular procedures without creating new information. The paper formalizes aperture as selective sensitivity and argues for multi-lens analysis with strict null controls to prevent pattern hunting.

1. Introduction

Algorithms are often described as recipes or procedures. That description is correct but incomplete. For discovery tasks, an algorithm can also be understood as a lens: it determines which relationships are emphasized, which are suppressed, and what counts as a detectable pattern.

The lens metaphor becomes especially useful when paired with an adjustable aperture. Analysts can alter resolution, feature weighting, representation, search depth, coordinate system, radix, tolerance, or objective function according to what they seek.

2. Information Does Not Change; Accessibility Can

An invertible transformation does not create new information. Yet it can radically alter the ease with which a particular procedure detects structure. Periodicity that is obscure in a time series can become obvious in a frequency representation. Multiplicative growth can become linear under a logarithm. A rational relationship can become short or terminating in one radix and repeating in another.

The distinction is between information content and information accessibility.

3. From Planetary Radix Tests to Observer-Centered Method

The motivating experiments initially asked whether planetary dynamics generated intrinsic preferred radix landscapes. Independent scoring and metric-neutral controls substantially weakened that strong interpretation. A later discovery-performance formulation produced a narrower result: changing radix could modestly alter recovery performance for some rational, resonant, or recurrence-like relationships while leaving robust spectral methods nearly unaffected.

The methodological lesson is more general than radix choice. The planet determines the relationships; the observer chooses the lens.

4. Defining Aperture

For an algorithmic lens, aperture denotes the set of structures to which the procedure is made sensitive. A narrow aperture may search only low-order rational relationships. A wider aperture may admit many candidate transformations but incur higher false-positive risk. Resolution corresponds to tolerance or granularity. Focus corresponds to the objective or relationship class.

This vocabulary makes algorithm design intuitive without changing its mathematics.

5. Search and Sort as Perspective Operations

Sorting is not neutral. Sorting by magnitude, frequency, correlation, spatial location, complexity, or recurrence produces different visible organizations of the same records. Search is similarly conditional: a query defines what can count as relevant.

An algorithm therefore operationalizes a question. Change the question and the same dataset becomes a different landscape of salience.

6. Multi-Lens Analysis

A disciplined exploratory system can intentionally examine a dataset through a preregistered portfolio of lenses: conventional physical coordinates, normalized ratios, frequency transforms, logarithmic transforms, alternative radix encodings, graph representations, and domain-specific invariants.

Candidate discoveries should then be returned to the original measurements and tested for predictive value. The transformation is a discovery aid, not evidence by itself.

7. The Multiple-Lens Hazard

Every additional lens increases the opportunity to find accidental structure. This is the algorithmic equivalent of trying enough filters until noise looks meaningful. The framework therefore requires matched nulls, held-out data, multiple-testing correction, and independent verification.

An adjustable aperture is useful only when the observer also records how far it was opened.

8. Human and Machine Lenses

Humans possess strong representational biases: familiar multiples, visual symmetry, linguistic categories, and learned notation. Algorithms possess different biases determined by architecture and objective. A representation that helps a human may not help a Fourier method; a digit-oriented procedure may respond strongly to radix while a representation-agnostic estimator does not.

The useful question is therefore never simply “Which representation is best?” It is “Best for which observer, algorithm, relationship, and task?”

9. Conclusion

An algorithm is usefully understood as a mathematical lens whose aperture can be adjusted to resolve different structures in invariant information. The metaphor is not merely pedagogical: it captures a real methodological fact about representation, inductive bias, search, and discoverability. The discipline lies in remembering that a clearer pattern through a lens is a candidate for investigation, not yet a fact about the world.

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