Carry the structure
into the computation.
How can a numerical representation preserve the relationships that make a measurement meaningful?
My approach grew from engineering work at NASA and the collaborations and independent studies that followed.
01 / Research question
A correct calculation can still mislead.
Computational systems collect measurements of biological processes, physical systems and human interactions. We analyze them through distances, averages, correlations and predictions. Those operations depend on assumptions about how the measurements relate to one another.
A direction wraps around. A physical configuration may have several equivalent descriptions. A numerically small change may be biologically significant. If a representation discards these relationships, a calculation can be correct on its own terms and still give a misleading account of the measured system.
My research asks how to carry the geometric structure of that system into the mathematics used to compute with it. The aim is to make the relationships part of the representation, the comparison and the transformation from the beginning.
For a measured example, see HELICITY’s LAMINAR study: a controlled replay keeps the trade events fixed while changing the numerical price scale. The classifier’s response changes, making the representation part of the measurement.
02 / Translations & geometry
What changes, and what stays the same?
I study transformations connecting different descriptions of the same phenomenon. I use “translator systems” for the rules or implementations that move between those descriptions, and ask what geometric structure they preserve.
A concrete example / Direction and orientation
A full turn for an arrow. A half turn for an axis.
A directed arrow in a plane returns to its original direction after a full turn: τ = 2π. An unoriented axis returns after π, because reversing it describes the same axis.
| Description | Same state | What it retains |
|---|---|---|
| Directed arrow | θ and θ + 2π | Orientation and which way the arrow points. |
| Unoriented axis | θ and θ + π | Orientation, with opposite directions identified. |
The map that forgets the arrowhead is a double cover: two opposite arrow directions correspond to each axis. Quantities unchanged by reversing the arrow can pass to the axis description; a distinction between forward and backward cannot.
This example illustrates the research question. It does not establish that a particular geometric model is appropriate for every dataset.
For a Riemannian model, the metric must describe what counts as meaningful local change. A fiber-bundle model can help organize a base description together with associated states or descriptions over it, provided the relevant spaces and their relationships are specified. Choosing either framework requires a domain-specific argument.
Explore two implementations of that question: ICARUS connects descriptions of a body and its surrounding air, while GIGI makes geometric relationships available through a database interface. For a construction you can manipulate directly, carry a frame around a loop and inspect the rotation it accumulates.
The research goal is to connect these mathematical descriptions to implemented translations: identify what survives, measure the distortion, and derive bounds on when that distortion can affect a decision. Those bounds are objectives to establish under stated assumptions, not a general guarantee already obtained.
03 / Research practice
From a representation to a testable claim.
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Curate the measurement.
Record what was measured, its units, collection conditions and provenance. Keep the context needed to interpret a value attached to it.
In USAspending / IDAS, each finding travels with its evidence bundle: configuration, schema snapshots, sampled rows and execution logs remain available for inspection. In OSHUN, experimental context is part of interpreting a permeability measurement.
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Specify identity and meaningful change.
State which transformations leave the phenomenon unchanged, which differences matter, and why a proposed space or distance fits the domain.
PRISM follows a payment across different message formats. Similar records are candidates for a match; deciding that they represent the same payment requires the transaction context and a separate matching decision.
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Make the translation explicit.
Describe the encoding, coordinate change or reduction. Identify what it preserves, what it merges and what information it discards. When an encoding changes dynamically, separate its effects from changes in the measured system.
AMC makes this the central engineering problem: adapt the telemetry encoding while tracking translations and lineage. The question is whether a changed vector reflects changed telemetry or a changed description.
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Connect distortion to a decision.
Define the error being measured and the decision it could affect. Seek a bound under explicit assumptions, or an empirical estimate with uncertainty when a bound is unavailable.
AMC-Edge specifies precision choices, error checks and fallback conditions. For a simpler decision rule, change the row counts in the USAspending example and see exactly when the configured parity threshold is crossed.
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Test, compare and revise.
Make a prediction, choose the comparison and failure criteria, then test against appropriate evidence. Distinguish proof, simulation and observation. Retain unsuccessful cases and correct the record when an argument or test fails.
Compare SSSP’s workload-dependent gains and slowdowns with LAMINAR’s real-input and surrogate controls. The comparison determines what a reported result can support.
This approach grew from Pandora’s curatorial practice and my work on changing telemetry representations at NASA. Its usefulness across domains is something to investigate separately in each setting.
04 / Reading the collection
What does a result establish?
| Record | What it provides | What it does not establish |
|---|---|---|
| Theoretical manuscript Example: flat models with connection |
An argument under stated hypotheses. | Independent proof verification or an empirical law. |
| Computational study Example: CHIHIRO geometry comparisons |
Results for a specified model, implementation and inputs. | Performance on every workload or validity in a physical system. |
| Technical report / artifact Example: USAspending check definitions |
An inspectable procedure, implementation or record. | Correctness merely because code or data are available. |
| Working-paper draft Example: DEMETER evaluation proposal |
A bounded question and proposed evaluation for review. | A completed experiment, accepted paper or new result. |
Attach conditions to comparisons.
Report the data, baseline, implementation, hardware, preprocessing, stopping criterion and uncertainty. Different metrics answer different questions. An algebraic identity recovered to floating-point precision is a consistency check; it is not an independent physical observation.
The CHIHIRO study keeps shape comparisons, equilibrium records and separate instability diagnostics visible, including a discrepancy between a saved summary and its underlying rows.
Keep negative results in view.
The SSSP companion records cases where bucket methods lose and lookahead brings no measured insertion benefit. OSHUN reports a dose-response test that did not support the intended inference. These outcomes help define the limits of the methods.
State mathematical claims precisely.
Separate definitions, conditional theorems and conjectures. A result under additional axioms does not settle a classical statement without a justified connection to its original hypotheses. Shared notation across domains does not make their quantities interchangeable.
For a concrete example, inspect the interpolation counterexample: an error bound that holds over an interval does not automatically hold from below at every point. The review identifies the precise step that fails.
Make corrections part of the record.
An error in a test or proposed proof can require a corrected method, a narrower conclusion or a retracted claim. Versions and supporting evidence should make that change inspectable. Existing deposits remain linked under their original titles; revised summaries do not silently rewrite them.
The cryptographic measurement account keeps withdrawn attack claims separate from later experiments and retained negative results. The mathematical review record likewise separates checked statements, counterexamples and remaining proof obligations.
The October 2026 catalog revision checked deposit metadata and reviewed site and project materials. It did not independently reproduce every experiment or validate every proposed proof. New working drafts are dated scope notes and evaluation protocols, with no new experiments performed for them. Companion applications retain their own histories and may contain earlier formulations.