---
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---
## Established results

The local two-field parent action, both variations, reciprocal sources and energy-square completion are established. The manuscript also gives an independent frame-stretch extension for the gravitational far field, the coupled vacuum spectrum and an analytic radius for the scale model. The task is joint closure of these existing ingredients, not finding an entirely unwritten master equation.

<Equation name="sd-coupling" />

<Equation name="sd-dual-action" />

## Research still required

Select complete variables, symmetries, boundaries and microscopic parameters. Use one action to test feedback-renormalized longitudinal/transverse propagation, localized matter, long-range gravitational sources and motion families. Local density response and independent stretch are distinct variables; speeds must be matched after order-parameter feedback.

## The next reproducible deliverable

Deliver the full variation, conserved quantities, vacuum spectrum and parameter domain of one explicit truncation; map it to the static far field, stable localization and accompanying-field envelope, recording incompatible conditions.

## Scope and research links

Reciprocal variation, an energy lower bound and a radial stationary point are distinct established results; none alone completes the particle spectrum, nonradial stability or all measurement responses.

[master equation](/en/theory/space-dynamics/master-equation) · [longitudinal transverse coupling](/en/theory/space-dynamics/longitudinal-transverse-coupling) · [longitudinal transverse coupling](/en/theory/space-dynamics/longitudinal-transverse-coupling) · [master equation](/en/theory/space-dynamics/master-equation) · [space dynamics](/en/theory/space-dynamics)

<CanonicalSource />