--- { "id": "ASD-GUIDE-EN-OVERVIEW-LEARNING-PATHS-MATHEMATICAL", "path": "/en/overview/learning-paths/mathematical", "title": "Mathematical Path", "title_en": "Mathematical Path", "category": "Research navigation", "summary": "Produce a checkable derivation with equation IDs, premises, retained orders, remainders and validity conditions.", "template": "document", "version": "1.0", "updated": "2026-09-28", "canonical": false, "source_tier": "website-guide", "theory_status": "FOUNDATION", "evidence_status": "DOCUMENT", "locale": "en", "translation_path": "/overview/learning-paths/mathematical", "depends_on": [], "related": [ "/en/overview/learning-paths", "/en/theory/metrology", "/en/theory/time/clocks-and-frequency", "/en/theory/lve/derivation", "/en/theory/lve/tests", "/en/theory/dynamics/momentum", "/en/theory/dynamics/energy", "/en/theory/dynamics/force-and-work", "/en/theory/gravity/light-deflection", "/en/theory/gravity/shapiro-delay", "/en/theory/gravity/orbital-dynamics", "/en/theory/space-dynamics/wave-propagation", "/en/theory/space-dynamics/continuum-dynamics", "/en/theory/space-dynamics/longitudinal-transverse-coupling", "/en/theory/quantum/effective-wave-equation", "/en/theory/quantum/uncertainty-probability", "/en/theory/quantum/detection-statistics" ], "open_problems": [], "equations": [], "sources": [ { "label": "Existing website content and Research Ledger; editorial reading navigation", "kind": "website-content" } ], "section_anchors": [ "step-definitions", "step-lve", "step-dynamics", "step-gravity", "step-continuum", "step-envelope" ], "translation_of": "/overview/learning-paths/mathematical", "translation_version": "1.0" } --- For readers who want to check premises, elimination, variation, approximations and limits in sequence. **Prerequisites** Vector calculus and ordinary differential equations; later steps use variational methods, Fourier analysis and linear partial differential equations. **Outcome** Produce a checkable derivation with equation IDs, premises, retained orders, remainders and validity conditions. The checkboxes below record your own reading on this device. Every source retains its own version, scope and scientific status. ## 1. Fix definitions and measurement maps Separate operational constraints from physical inputs used to close spatial relations. Read in this order: [Metrological Foundations](/en/theory/metrology) → [Time · Clocks and Frequency](/en/theory/time/clocks-and-frequency). **Checkpoint** Which equalities are definitions, and which require physical constitutive relations? **Next action** Build a symbol table with units, reference conditions and the final readings to be compared. [Discuss this source in Researcher](/researcher?mode=critique&context=%2Fen%2Ftheory%2Fmetrology&lang=en). ## 2. Check LVE step by step Check path relations, elimination and review conditions in the original derivation. Read in this order: [Length Vector Effect · Full Derivation](/en/theory/lve/derivation) → [Experimental Relations and Tests](/en/theory/lve/tests). **Checkpoint** Does eliminating a variable require an additional independence or configuration condition? **Next action** Open an equation detail from the article and record its upstream dependencies and one limiting-case check. [Discuss this source in Researcher](/researcher?mode=critique&context=%2Fen%2Ftheory%2Flve%2Fderivation&lang=en). ## 3. From action to dynamical quantities Check momentum, energy, force and work using the existing dynamical relations. Read in this order: [Dynamics · Inertia and Momentum](/en/theory/dynamics/momentum) → [Dynamics · Energy and Action](/en/theory/dynamics/energy) → [Dynamics · Force and Work](/en/theory/dynamics/force-and-work). **Checkpoint** Which variables and reference conditions are held fixed when derivatives are taken? **Next action** Independently recalculate one low-speed limit and separate source steps from your added calculations. [Discuss this source in Researcher](/researcher?mode=critique&context=%2Fen%2Ftheory%2Fdynamics%2Fmomentum&lang=en). ## 4. Check rays and orbits in a background Distinguish reference length, physical spatial length, optical-path weighting and action. Read in this order: [Light Deflection · From Optical Path to Rays](/en/theory/gravity/light-deflection) → [Shapiro Delay · Excess Optical Path and Propagation Counts](/en/theory/gravity/shapiro-delay) → [Orbital Dynamics · Action, Conservation and Precession](/en/theory/gravity/orbital-dynamics). **Checkpoint** Is a coordinate conversion counted twice in the optical path? Are endpoints and approximation orders consistent? **Next action** Check one integral’s endpoints, dimensions and weak-field limit, retaining the stated status of higher-order differences. [Discuss this source in Researcher](/researcher?mode=critique&context=%2Fen%2Ftheory%2Fgravity%2Flight-deflection&lang=en). ## 5. Decompose propagation sectors Check mode decomposition, propagation speeds and required constitutive conditions from continuum degrees of freedom. Read in this order: [Wave Propagation and the Equal-Speed Condition](/en/theory/space-dynamics/wave-propagation) → [Continuum Dynamics](/en/theory/space-dynamics/continuum-dynamics) → [Longitudinal–Transverse Coupling](/en/theory/space-dynamics/longitudinal-transverse-coupling). **Checkpoint** Under which conditions is equal longitudinal/transverse speed obtained, and what closure relations are still missing for coupling? **Next action** Separate established linear results from candidate coupling terms; do not use candidates to claim a closed master equation. [Discuss this source in Researcher](/researcher?mode=critique&context=%2Fen%2Ftheory%2Fspace-dynamics%2Fwave-propagation&lang=en). ## 6. Check the envelope and statistical boundary Keep the exact remainder after carrier substitution, then check approximation, matching and conditional detection relations. Read in this order: [How Does a Spatial Field Yield an Effective Wave Equation?](/en/theory/quantum/effective-wave-equation) → [Uncertainty: Fourier Width Bounds for Wave Packets](/en/theory/quantum/uncertainty-probability) → [How Can Continuous Fields Yield Discrete Detection Statistics?](/en/theory/quantum/detection-statistics). **Checkpoint** What scale conditions justify dropping a term? What is needed to turn Fourier width into momentum width? **Next action** Export a contextual equation task and use Critique mode to examine remainders and validity; retain human review of the result. [Discuss this source in Researcher](/researcher?mode=critique&context=%2Fen%2Ftheory%2Fquantum%2Feffective-wave-equation&lang=en). [Return to all learning paths](/en/overview/learning-paths)