Mathematician / Theoretical Physicist (SME — Delay Calculus)

About the Role

We’re building a co-timing engine: a platform that treats multiple measurement paths as “the same process up to delay (and sometimes gain),” aligns them onto a shared timebase, cancels what is truly shared, and turns what survives into an observable tied to geometry and medium.

At the heart of this is a custom delay calculus: an operator algebra over time delays (delay bricks, delay polynomials, composition rules, window validity conditions, commutation failures, stability guarantees). We need a domain expert who can own the mathematical core—keep it honest, extend it, and translate it into rigorous, testable contracts engineering can implement.

This is not a “write papers only” role. You are the SME who makes the math shippable.

What You’ll Own

• The delay calculus core: formalize and extend the operator algebra (operators, compositions, equivalence, invariants).
• Conditions of validity (“when the math applies”): define window-based assumptions (slow drift, commutation bounds, stationarity/coherence) and how violations should surface.
• Stability & identifiability guarantees: characterize when estimation/cancellation is well-posed; define stability criteria and no-go boundaries.
• Receipts & contracts: turn theoretical statements into measurable conditions, acceptance tests, and refuse-to-claim criteria.
• Bridge to implementations: ensure the math maps cleanly to estimators, diagnostics, and APIs (time-domain and spectral-domain).
• Research roadmap: identify high-leverage gaps (noncommuting delays, time-varying composition, multipath ambiguity) and propose tractable solutions.

What You’ll Do

• Derive and validate operator identities and composition rules for delay polynomials.
• Provide clean if/then structure: if these window promises hold, then these cancellations/estimators are valid; else expect these residual signatures.
• Develop proofs, proof sketches, and sanity checks that survive finite windows, discretization, and noise.
• Co-design diagnostics with the team: commutator residuals, stability covers, validity scores, failure-mode labels.
• Create reference constructions and counterexamples that become QA datasets and regression tests.
• Review algorithm designs for mathematical correctness and boundary-case behavior.
• Write internal notes and external-facing explanations (when appropriate) that translate deep math into operational language.

Concrete Deliverables

• A Delay Calculus Spec: definitions, operator rules, composition identities, and conventions (units, normalization, window semantics).
• A Validity & Failure Taxonomy: assumptions → tests → expected signatures → mitigation.
• A theorem-to-test mapping: each key claim paired with measurable metrics and automated acceptance tests.
• Reference constructions: toy benches/synthetic datasets illustrating success and failure modes.
• A math review gate for changes to the core calculus (so foundations don’t drift into folklore).

Required Qualifications

• Advanced training in mathematics, theoretical physics, or a closely related field (PhD preferred; equivalent depth acceptable).
• Strong command of linear systems, operator methods, Fourier/spectral reasoning, and rigorous reasoning under approximation.
• Comfort translating between symbolic reasoning and numerical reality: finite windows, noise, discretization, identifiability limits.
• Ability to communicate clearly with engineers: definitions, assumptions, invariants, and “what breaks” explanations.

Preferred Qualifications

• Prior exposure to time-delay interferometry (TDI), delay operators, or related synchronization/cancellation frameworks.
• Background in estimation theory, inverse problems, or applied harmonic analysis.
• Experience collaborating with software teams (writing specs, defining tests, reviewing implementations).
• Familiarity with proof-adjacent workflows (not required, but a plus).

How You’ll Be Measured (First 60–90 Days)

• You can explain the delay calculus precisely and identify sharp assumptions and boundaries.
• You deliver at least one high-leverage contract (validity condition + diagnostic + acceptance test) that measurably reduces false confidence.
• You produce reference examples/counterexamples that become part of the QA suite.
• Engineering teams implement features faster because the math is explicit, stable, and testable.

Working Style

• You treat “it seems to work” as a hypothesis, not a conclusion.
• You like clean definitions and explicit assumptions, especially where systems can silently fail.
• You enjoy turning deep structure into practical guardrails: “valid” / “do not trust—here’s why.”

Title & Level

Mathematician / Theoretical Physicist (SME — Delay Calculus) (Senior/Principal-level technical authority), partnering tightly with applied math/DSP, systems, and validation teams.