What This Is
Quantitative Unique Continuation as a Rigidity Principle
This archive contains a complete taxonomy classifying scale-critical PDEs by whether quantitative unique continuation (UC) serves as a rigidity principle that excludes finite-time singularities.
Key Results
- Universal UC Pipeline: Six-step pipeline (Carleman, three-cylinder/cone, doubling, frequency, vanishing order, blow-up contradiction) works for elliptic, parabolic, and hyperbolic (fixed geometry) operators.
- Form-Boundedness as Universal Coefficient Class: Form-boundedness is the invariant that makes UC work across all operator types.
- Rigidity Theorem: UC + form-boundedness ⇒ no finite-time singularities for systems in rigidity class $\mathcal{R}$.
- Obstruction Theorem: Six geometric obstructions prevent UC closure for fully dynamical Lorentzian geometries (Einstein vacuum).
- Taxonomy Theorem: Every scale-critical PDE falls into either rigidity class $\mathcal{R}$ or obstruction class $\mathcal{O}$.
Status Taxonomy
- CLOSED: Proof complete, all dependencies verified
- CONDITIONAL: Proof complete but depends on external validation
- FORMULATION: Formulation study, no claims
- OBSTRUCTED: Geometric obstructions identified, framework does not close
Status Taxonomy
Rigidity Class $\mathcal{R}$ (CLOSED)
| Class | Examples | Status |
|---|---|---|
| $\mathcal{R}_{\mathrm{ell}}$ (Elliptic) | Schrödinger | CLOSED |
| $\mathcal{R}_{\mathrm{par}}$ (Parabolic) | NS, MHD, YM heat, HM heat | CLOSED |
| $\mathcal{R}_{\mathrm{hyp}}$ (Hyperbolic fixed) | Wave maps, YM wave | CLOSED |
Obstruction Class $\mathcal{O}$ (OBSTRUCTED)
| Class | Examples | Status |
|---|---|---|
| $\mathcal{O}_{\mathrm{dyn}}$ (Dynamical hyperbolic) | Einstein vacuum | OBSTRUCTED |
The Boundary
The boundary between $\mathcal{R}$ and $\mathcal{O}$ is geometric:
- Fixed geometry: Minkowski space ⇒ $\mathcal{R}_{\mathrm{hyp}}$
- Dynamical geometry: Curved spacetime ⇒ $\mathcal{O}_{\mathrm{dyn}}$
Papers
Rigidity Class $\mathcal{R}$
Obstruction Class $\mathcal{O}$
Taxonomy
UC Rigidity Taxonomy: A Taxonomy of Success and Obstruction
Status: CLOSED
PDF | Expert Packet | Source
Verify
How to Reproduce
See VERIFICATION.md for detailed instructions.
How to Replay
- Clone the repository
- Checkout RELEASE_v1.0 tag
- Verify PDF hashes
- Rebuild all papers
- Verify dependency chain
Hashes
All PDFs are hashed with SHA-256. See HASHES.md for verification.
Engine
These artifacts were compiled with the Convergence Engine, an epistemic compiler that finds irreducible laws of nature.
Domain Strategy
- convergenceengine.dev: The epistemic compiler, passes, IR, replay, attack harness. Audience: engineers, formal methods, AI, verification.
- invariantinstitute.org: A neutral, institutional-sounding archive of rigidity results and proofs. Audience: mathematicians, physicists, scientists.