Registration Geometry
What exact geometric structure can be built directly from the registration model, without assuming a background spacetime?
The five-paper completion spine (registration tensor → coordinate-free geometry → Lorentzian signature → physical metric candidate → field-to-metric map) plus the canonical ACRF-4 1 ⊕ 3 decomposition.
Established
- ACRF-4 admits an exact 1 ⊕ 3 decomposition of its registration coordinates.
- A coordinate-free registration tensor, a registration-signature form, and a normalized physical-metric candidate have each been defined and connected by an admissible field-to-metric map.
Open problems
Metric and continuum reconstruction
A completed, general theorem reconstructing a physical metric from registration structure.
Related threads: Metric Reconstruction, Continuum Limits
Sources: Registration Geometry I: Stable Records and a Memory-Sensitive Effective Metric on a Supplied Lorentzian Background, Registration Geometry V: Induced Einstein Dynamics from a Retained Registration Spectrum
Relevant claims
ACRF-4 admits an exact 1 ⊕ 3 decomposition
R⁴ decomposes exactly as U ⊕ H4, with U = span{1} of dimension 1 and H4 = { r : 1ᵀr = 0 } of dimension 3, and R⁴ / U ≅ H4 ≅ R³.
Scope: A structural linear-algebra fact about the canonical finite ACRF-4 registration model. It does not by itself identify the four ACRF-4 coordinates with physical quantities.
- Does not constitute a derivation of physical spacetime from ACRF-4.
- Host calibration is downstream and does not supply a preferred basis automatically.
Last reviewed 2026-08-27
Relevant publications
The Registration Tensor: From Measure-Valued Records to a Coordinate-Covariant Field
Paper 1 of the five-paper completion spine: defines the registration tensor object.
Lorentzian Signature from Registration Order and Transverse Record Geometry
Paper 3 of the five-paper completion spine: introduces a registration-signature form.
The Physical Metric as Normalized Registration Geometry
Paper 4 of the five-paper completion spine: defines a physical normalization map.
From Registration Field Aμν to Physical Metric gμν
Paper 5 (final) of the five-paper completion spine: an admissible field-to-metric map.